9th Grade Mathematics Worksheets - Free Printable

Let’s solve each problem one by one, step by step. I’ll make sure to check my work carefully before giving the final answer.

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(1) ABCD is a quadrilateral in which AB is parallel to CD and AD = BC = 5 cm. If ∠A = 105°, find the value of ∠B.

Since AB is parallel to CD, and AD = BC, this is an isosceles trapezoid (non-parallel sides are equal).

In a trapezoid with AB || CD, consecutive angles between the same leg are supplementary (add up to 180°).

So, ∠A + D = 180°
→ ∠D = 180° - 105° = 75°

But we need ∠B.

In an isosceles trapezoid, base angles are equal:
∠A = ∠B? No — actually, since AB is the top base and CD is the bottom base, then:

Actually, let’s think again.

If AB || CD, then:

- ∠A and ∠D are on the same side → they are adjacent angles along leg AD → so they add to 180°.
- Similarly, ∠B and ∠C add to 180°.

Also, because it’s isosceles (AD = BC), the base angles are equal:

→ ∠A = ∠B? Not necessarily. Actually, in an isosceles trapezoid, the angles adjacent to each base are equal.

Wait — better way:

In isosceles trapezoid ABCD with AB || CD and AD = BC:

Then ∠A = ∠B? Let me draw mentally:

Points: A -- B (top base)
D -- C (bottom base)

Legs: AD and BC.

Since AB || CD, then:

∠A + D = 180°
∠B + ∠C = 180°

And because it’s isosceles, ∠A = ∠B? No — actually, ∠A = ∠B only if it’s symmetric that way.

Actually, standard property: In isosceles trapezoid, base angles are equal. So angles at A and B are NOT both base angles — depends on which base.

Actually, if AB is the top base, then angles at A and B are the “top” angles, and angles at D and C are the “bottom” angles.

In isosceles trapezoid, the two angles adjacent to each leg are equal? No.

Correct property: The base angles are equal. That means:

∠A = B? Only if AB is the base and it’s symmetric — but actually, no.

Standard definition: In isosceles trapezoid, the angles adjacent to each base are equal.

So if AB and CD are the bases, then:

∠A = ∠B? No — that would be if it were a rectangle.

Actually, correct: In isosceles trapezoid, the angles on the same base are equal.

So for base AB: ∠A and B should be equal? But that can’t be if AB is parallel to CD and legs are slanted.

I think I have it backwards.

Let me recall: In trapezoid ABCD with AB || CD, and AD = BC (legs equal), then it is isosceles, and the base angles are equal meaning:

∠A = B? No — actually, ∠A = ∠D? No.

Better: The angles adjacent to each leg are supplementary, and the base angles are equal across.

Actually, standard fact: In isosceles trapezoid, the angles adjacent to the same base are equal.

So for base AB: ∠A and B are not both on the same base in terms of being "base angles" — wait.

Perhaps it's easier: Since AB || CD, then ∠A + ∠D = 180°, and ∠B + ∠C = 180°.

Also, because AD = BC, the trapezoid is symmetric, so ∠A = ∠B? Let's test with numbers.

Suppose ∠A = 105°, then since AB || CD, ∠D = 180° - 105° = 75°.

By symmetry (since AD = BC), the other pair: ∠B should equal ∠A? Or ∠B should equal ∠C?

Actually, in isosceles trapezoid, the base angles are equal, meaning the two angles at the top base are equal, and the two at the bottom are equal.

So if AB is the top base, then ∠A and ∠B are the top angles, and they should be equal? But that would mean ∠A = B = 105°, then ∠D = C = 75°, and sum is 105+105+75+75=360°, which works.

Is that possible? In a trapezoid with AB || CD, if ∠A = ∠B, then the legs must be such that it's symmetric.

Yes, that's correct for isosceles trapezoid: the base angles are equal. So angles at A and B are both adjacent to base AB, so they are equal.

Similarly, angles at C and D are equal.

So if ∠A = 105°, then ∠B = 105°.

But let me confirm with the parallel lines.

If AB || CD, and we consider leg AD, then ∠A and D are consecutive interior angles, so they add to 180°.

Similarly, for leg BC, ∠B and ∠C add to 180°.

Now, if the trapezoid is isosceles, then ∠A = ∠B, and ∠D = ∠C.

Given ∠A = 105°, then ∠B = 105°, ∠D = 75°, ∠C = 75°.

Sum: 105+105+75+75=360°, good.

And AB || CD is satisfied.

So answer is 105°.

But the question says "find the value of ∠B", and given ∠A=105°, so ∠B=105°.

I think that's it.

But let me double-check online or standard knowledge.

Upon second thought, in some definitions, the base angles refer to the angles at each base, so for base AB, angles at A and B are the base angles, and they are equal in isosceles trapezoid.

Yes.

So ∠B = 105°.

But I recall that sometimes people get confused.

Another way: the non-parallel sides are equal, so the base angles are equal. Specifically, angle at A equals angle at B? No.

Let's label properly.

Assume points in order: A, B on top, D, C on bottom, so AB top, DC bottom, AD left leg, BC right leg.

Then, angle at A is between AB and AD, angle at B is between AB and BC.

In isosceles trapezoid, angle at A should equal angle at B? Only if it's symmetric, which it is.

Actually, no: in standard isosceles trapezoid, the angles adjacent to each leg are not necessarily equal, but the angles on the same base are equal.

For example, angle at A and angle at B are both on the top base AB, so they should be equal.

Yes, that's correct.

To confirm, suppose it's a rectangle: all angles 90°, so yes.

If it's not rectangle, say top base shorter, then angles at A and B are obtuse if bottom is longer, and equal.

In this case, ∠A=105°, so ∠B=105°.

But let's see the sum: if ∠A=105°, and AB||CD, then ∠D=75°.

By symmetry, since AD=BC, the figure is symmetric, so ∠B should be equal to ∠A, and ∠C equal to ∠D.

Yes.

So answer is 105°.

But I feel like maybe I'm missing something because usually in trapezoids, consecutive angles are supplementary, but here ∠A and B are not consecutive in that sense; they are on the same side.

Consecutive angles sharing a leg are supplementary, but ∠A and B share the base AB, not a leg.

Angles sharing a leg: ∠A and D share leg AD, so they are supplementary.

∠B and ∠C share leg BC, so supplementary.

∠A and B do not share a leg; they are adjacent on the base.

In isosceles trapezoid, ∠A = ∠B.

Yes, I think it's correct.

So for (1), ∠B = 105°.

But let me calculate numerically.

Suppose we drop perpendiculars from A and B to CD, meeting at P and Q.

Then, since AD=BC=5, and angles, but perhaps overcomplicating.

I think it's fine.

So answer for (1) is 105 degrees.

But the problem says "find the value of ∠B", and given ∠A=105°, so likely 105°.

Perhaps they mean something else, but I think it's correct.

Moving on, and I'll come back if needed.

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(2) A sphere is just enclosed inside a cube of volume 72 cm³. Find the volume of the sphere.

"Just enclosed" means the sphere is inscribed in the cube, so it touches all faces, so diameter of sphere equals side of cube.

Volume of cube = s³ = 72 cm³, so s = ∛72.

Simplify: 72 = 8 * 9 = 2^3 * 3^2, so s = ∛(8*9) = 2 ∛9.

Diameter of sphere = s, so radius r = s/2 = ∛9.

Volume of sphere = (4/3)πr³ = (4/3)π (∛9)^3 = (4/3)π * 9 = 12π.

Is that right?

r = s/2, s³ = 72, so r³ = (s/2)^3 = s³ / 8 = 72 / 8 = 9.

Oh! So r³ = 9.

Thus volume = (4/3)π r³ = (4/3)π * 9 = 12π.

So volume is 12π cm³.

We can leave it as 12π, or approximate, but since no specification, exact form is fine.

So answer is 12π.

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(3) Vinayak got an average score of 82.25 in 4 tests. He got 83 as the average of the highest 3 scores, and his lowest two scores are the same numbers. What is the average of his highest two scores?

Let the four scores be a, b, c, d, with a ≤ b ≤ c ≤ d.

Given:

Average of all four: (a+b+c+d)/4 = 82.25 → sum = 82.25 * 4 = 329.

Average of highest three: (b+c+d)/3 = 83 → sum b+c+d = 83 * 3 = 249.

Lowest two scores are the same: so a = b. (since a and b are the two lowest, and they are equal)

The problem says "his lowest two scores are the same numbers", so yes, a = b.

Now, from total sum: a + b + c + d = 329

But a = b, and b + c + d = 249

So substitute: a + (b + c + d) = 329 → a + 249 = 329 → a = 329 - 249 = 80.

Since a = b, b = 80.

Now, b + c + d = 249 → 80 + c + d = 249 → c + d = 249 - 80 = 169.

Now, we need the average of the highest two scores, which are c and d.

So average = (c + d)/2 = 169 / 2 = 84.5.

Is that it?

Let me verify.

Scores: a=80, b=80, c and d sum to 169.

Total sum: 80+80+169=329, yes.

Highest three: b,c,d =80 + c + d =80+169=249, average 83, yes.

Lowest two are both 80, same.

Average of highest two: (c+d)/2=169/2=84.5.

So answer is 84.5.

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(4) Examine if the following numbers are rational or irrational.

Recall: rational number can be written as p/q where p,q integers, q≠0. Irrational cannot.

A) √6 — 6 is not perfect square, so irrational.

B) √5 - √7 — both irrational, and their difference: suppose it were rational, say r, then √5 - √7 = r, so √5 = r + √7, square both sides: 5 = r² + 2r√7 + 7 → 5 - 7 - r² = 2r√7 → -2 - r² = 2r√7. Left side rational, right side irrational unless r=0, but r=0 not possible. So irrational.

C) √5 — irrational, as 5 not perfect square.

D) (3 - √6)(3 - √6) — that's (3 - √6)^2 = 9 - 6√6 + 6 = 15 - 6√6. Now, √6 irrational, so this is irrational.

E) √5 - 7 — same as B, irrational.

F) √6 - 3 — √6 irrational, minus 3 still irrational.

All seem irrational? But let me check D again.

D is written as (3−√6)(3−√6), which is indeed (3−√6)^2.

But perhaps it's a typo? Or maybe it's meant to be something else, but as written.

In the image, it's D) (3−√6)(3−√6), so yes.

But let me compute numerically to confirm.

√6 ≈ 2.45, so 3 - 2.45 = 0.55, squared is about 0.3025, but exactly 15 - 6√6, which is irrational.

Similarly, others.

But is there any rational? For example, if it were √4, but no.

Perhaps F: √6 - 3, still irrational.

But let me see option E is same as B.

In the list:

A) √6 — irrational

B) √5 - √7 — irrational

C) √5 — irrational

D) (3−√6)(3−√6) = (3−√6)^2 = 9 - 6√6 + 6 = 15 - 6√6 — irrational

E) √5 - √7 — same as B, irrational

F) √6 - 3 — irrational

All are irrational? But perhaps I missed something.

Maybe D is meant to be (3−√6)(3+√6)? Because that would be difference of squares.

In the text, it's written as "(3−√6)(3−√6)", but perhaps it's a typo, and it should be (3−√6)(3+√6).

Because otherwise, all are irrational, but maybe the problem expects some rational.

Looking back at the user input: "D) (3−√6)(3−√6)" — it says twice minus.

But in many problems, it's often (a-b)(a+b).

Perhaps it's correct as is.

But let me assume it's as written.

So all are irrational.

But for the answer, probably we need to state for each.

The question is to examine, so perhaps list which are rational or irrational.

But since all seem irrational, but let me double-check D.

(3−√6)(3−√6) = 3*3 + 3*(-√6) + (-√6)*3 + (-√6)*(-√6) = 9 -3√6 -3√6 + 6 = 15 - 6√6, yes.

Now, 15 - 6√6, since √6 irrational, this is irrational.

Similarly, others.

But perhaps F: √6 - 3, same thing.

Unless one of them simplifies.

Another thought: is √5 - √7 rational? No, as I showed earlier.

Perhaps C is √4, but no, it's √5.

I think all are irrational.

But let's see the options; perhaps for D, if it were (3-√6)(3+√6) = 9 - 6 = 3, rational.

Probably a typo in the problem or in my reading.

In the user input: "D) (3−√6)(3−√6)" — it has two minuses.

But in the original image, it might be different, but since I don't have image, I have to go by text.

Perhaps it's (3−√6) times something else.

Another idea: perhaps it's (3−√6) and then another expression, but it's written as product.

I think I should proceed as is.

So for (4), all are irrational.

But to be precise, let's list:

A) irrational

B) irrational

C) irrational

D) irrational

E) irrational

F) irrational

But that seems odd for a problem.

Perhaps D is (3−√6)(3+√6), which is common.

In many textbooks, it's given as such.

Perhaps in the image, it's plus.

Since the user wrote " (3−√6)(3−√6) ", but maybe it's a mistake.

To be safe, I'll assume it's as written, but for the sake of the problem, perhaps it's intended to be difference of squares.

Let me check online or think.

Another way: perhaps " (3−√6)(3−√6) " is meant to be the square, but still irrational.

I think I'll note that if it's (3-√6)(3+√6), then it's 9-6=3, rational.

Otherwise, irrational.

But since the problem says "examine", and for accuracy, I'll say as written.

But for now, let's move on, and perhaps come back.

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(5) The length and breadth of a rectangle is 20 cm and 99 cm. Find the radius of circumcircle of this rectangle.

For a rectangle, the circumcircle is the circle passing through all four vertices. The diagonal of the rectangle is the diameter of the circumcircle.

Because in a rectangle, diagonals are equal and bisect each other, and the center is intersection of diagonals, and distance to vertices is half diagonal.

So, diagonal d = √(length² + breadth²) = √(20² + 99²) = √(400 + 9801) = √10201.

Now, what is √10201? Let me calculate.

100^2=10000, 101^2=10201, yes! Because 100^2=10000, 101^2=10000 + 200 +1=10201.

So d = 101 cm.

Thus, radius r = d/2 = 101/2 = 50.5 cm.

So answer is 50.5 cm.

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Now multiple choice.

(6) The three angles of a quadrilateral are 29°, 31° and 144° respectively. Find the fourth angle.

Sum of angles in quadrilateral is 360°.

So fourth angle = 360 - (29 + 31 + 144) = 360 - (60 + 144) = 360 - 204 = 156°.

Options: a.156° b.204° c.24° d.66°

So a.156°

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(7) Harsh is part of the school cricket team, and this year he has scored an average of 46 runs. He has played 5 innings so far, and his scores in 4 of them are 46, 45, 39, 55. What was his score in the last one?

Average of 5 innings is 46, so total runs = 46 * 5 = 230.

Sum of known four scores: 46 + 45 + 39 + 55.

Calculate: 46+45=91, 39+55=94, total 91+94=185.

So last score = 230 - 185 = 45.

Options: a.49 b.45 c.39 d.44

So b.45

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(8) From a point in the interior of an equilateral triangle, perpendiculars are drawn on the three sides. The lengths of the perpendiculars are 12 cm, 16 cm and 10 cm. Find the area of the triangle.

This is a standard problem. In an equilateral triangle, for any interior point, the sum of the perpendiculars to the three sides is constant and equal to the height of the triangle.

Is that true?

Yes, it's Viviani's theorem.

Viviani's theorem states that for any point inside an equilateral triangle, the sum of the perpendicular distances to the three sides is equal to the altitude of the triangle.

So, here, sum = 12 + 16 + 10 = 38 cm.

So altitude h = 38 cm.

Now, for equilateral triangle, area = (√3 / 4) * side², but also area = (1/2) * base * height.

Let side be s.

Height h = (√3 / 2) * s.

So, h = (√3 / 2) s = 38

Thus, s = 38 * 2 / √3 = 76 / √3

Rationalize: s = 76 √3 / 3

Now area = (1/2) * base * height = (1/2) * s * h = (1/2) * (76 / √3) * 38

Since h=38, and area = (1/2) * s * h, but s = 2h / √3, from h = (√3 / 2) s, so s = 2h / √3

Thus area = (1/2) * (2h / √3) * h = (h²) / √3

From h = (√3 / 2) s, so s = 2h / √3

Area = (√3 / 4) s² = (√3 / 4) * (4h² / 3) = (√3 / 4) * (4/3) h² = (√3 / 3) h² = h² / √3

(√3 / 3) = 1/√3, yes.

So area = h² / √3

But usually we rationalize.

h=38, so area = (38)^2 / √3 = 1444 / √3

Rationalize: (1444 √3) / 3

Now, numerically, but options are given.

Options:

a. 3.8 √3 cm² — too small

b. 833.69 cm²

c. 1083.797 cm²

d. 38 cm² — too small

Now, compute (1444 √3) / 3

√3 ≈ 1.732

So 1444 * 1.732 ≈ let's calculate: 1400*1.732=2424.8, 44*1.732≈76.208, total ≈ 2501.008

Divide by 3: ≈ 833.669, so about 833.67 cm²

Option b is 833.69 cm², close enough, probably rounding.

Exactly: 1444 / 3 * √3 = (1444 √3)/3

But option b is numerical, so likely b.

We can write area = (h² √3) / 3 ? No.

From earlier: area = (√3 / 4) s², and s = 2h / √3, so s² = 4h² / 3

Area = (√3 / 4) * (4h² / 3) = (√3 h²) / 3

Yes, so area = (√3 / 3) h² = h² / √3, same as before.

(√3 / 3) h² = (1/√3) h², but better to write as \frac{\sqrt{3}}{3} h^2

With h=38, area = \frac{\sqrt{3}}{3} * 1444 = \frac{1444 \sqrt{3}}{3}

Numerically, 1444 / 3 ≈ 481.333, times √3 ≈ 481.333 * 1.732 ≈ let's compute: 480*1.732=831.36, 1.333*1.732≈2.308, total ≈833.668, so 833.67, and option b is 833.69, probably due to √3 approximation.

Perhaps they used √3=1.73205 or something.

But clearly b is correct.

Option a is 3.8√3, which is small.

c is larger, d is 38, too small.

So b.

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(9) The average of 9 numbers is 22. If the average of first 4 results is 25 and that of last 4 is 21, then find the 4th number.

Total sum of 9 numbers: 9 * 22 = 198.

Sum of first 4: 4 * 25 = 100.

Sum of last 4: 4 * 21 = 84.

Note that the 4th number is included in both first 4 and last 4? No.

First 4: positions 1,2,3,4

Last 4: positions 6,7,8,9? Or 5,6,7,8? Typically, "last 4" means the last four, so if 9 numbers, last 4 are 6,7,8,9? No.

Standard: for 9 numbers, indices 1 to 9.

First 4: 1,2,3,4

Last 4: 6,7,8,9? But that skips 5.

Usually, "last 4" means the final four, so 6,7,8,9? But then number 5 is missing.

The 4th number is position 4.

The sets: first 4: 1,2,3,4

Last 4: probably 6,7,8,9? But then position 5 is not included in either.

But the total sum includes all.

Sum of first 4 + sum of last 4 = sum of 1,2,3,4,6,7,8,9 = total sum minus the 5th number.

But we need the 4th number.

The overlap: the first 4 and last 4 may overlap if "last 4" includes 4,5,6,7 or something.

Typically, in such problems, "first 4" means positions 1-4, "last 4" means positions 6-9, so position 5 is the middle one.

But here it asks for the 4th number, which is in the first 4.

Sum of first 4: 100

Sum of last 4: 84

But these two sets are disjoint if last 4 are 6,7,8,9.

Then sum of first 4 and last 4 is 100 + 84 = 184, which includes numbers 1,2,3,4,6,7,8,9.

Total sum is 198, so the missing number is position 5: 198 - 184 = 14.

But the question asks for the 4th number, not the 5th.

That doesn't help directly.

Perhaps "last 4" means positions 5,6,7,8? But then it overlaps with first 4 at position 4? No, first 4 is 1-4, last 4 if 5-8, then no overlap, but position 9 is missing.

Still, sum first 4 + last 4 = 1,2,3,4,5,6,7,8 = sum minus position 9.

Total sum 198, sum first 4=100, sum last 4=84, so sum of 1-8 = 100+84=184, so position 9 = 198-184=14.

But again, not the 4th.

The 4th number is included in the first 4, but we don't have direct info.

Perhaps "the last 4" includes the 4th? Unlikely.

Another interpretation: sometimes "first 4" and "last 4" overlap if the total is small, but for 9 numbers, first 4:1-4, last 4:6-9, so no overlap, and 5th is separate.

But we need 4th, which is in first 4.

From the sums, we have sum of 1-4 =100, but we don't know individual values.

The key is that the 4th number is counted in the first 4, but not in the last 4, assuming last 4 are 6-9.

But we have no equation for it alone.

Perhaps the "last 4" means the last four, which for 9 numbers could be 6,7,8,9, but then the 4th is not in it.

But the problem is to find the 4th number, so probably there is overlap or something.

Let me read: "the average of first 4 results is 25 and that of last 4 is 21"

And "find the 4th number".

In some contexts, "last 4" might mean positions 5,6,7,8, but then position 9 is excluded, and position 4 is not in last 4.

Still.

Perhaps the sets overlap at the 4th and 5th or something.

Another thought: for 9 numbers, the first 4 are 1,2,3,4; the last 4 are 6,7,8,9; but then the 5th is alone, and 4th is in first 4.

But to find 4th, we need more.

Unless the "last 4" includes the 4th, but that doesn't make sense.

Perhaps "last 4" means the final four, which are 6,7,8,9, but then the 4th is not related directly.

I recall that in such problems, when they say "first k" and "last m", and ask for a specific number, often there is overlap.

For example, if they said average of first 5 and last 5, then the 5th is in both.

Here, first 4 and last 4, for 9 numbers, if last 4 are 6,7,8,9, no overlap.

But perhaps "last 4" means 5,6,7,8? Then first 4:1-4, last 4:5-8, so they are adjacent, no overlap, and position 9 is missing.

Sum first 4 + last 4 = 1-8 = 100 + 84 = 184, total sum 198, so position 9 = 14.

But still not 4th.

The 4th number is part of the first 4, but we don't know which one.

Unless the problem is to find the 5th, but it says 4th.

Let me check the options: a.13 b.15 c.16 d.14

14 is an option, and we got position 9=14, but not 4th.

Perhaps "the 4th number" means the fourth in order, but in the context, it might be the overlapping one.

Another interpretation: perhaps "first 4" and "last 4" share the 4th and 5th or something, but for 9 numbers, if first 4 are 1,2,3,4, last 4 are 4,5,6,7? But then it's not "last 4"; last 4 should be the end.

Typically, "last 4" means the final four, so for n=9, positions 6,7,8,9.

But then no overlap with first 4.

Perhaps the total is 9, first 4:1-4, last 4:6-9, and the 5th is the middle, but the 4th is in first 4.

But to find it, we need more information.

Unless the "last 4" includes the 4th, but that doesn't make sense.

Let's think differently.

Let the numbers be x1 to x9.

Sum x1 to x9 = 9*22 = 198.

Sum x1 to x4 = 4*25 = 100.

Sum of last 4: if last 4 are x6,x7,x8,x9, sum = 4*21 = 84.

Then sum x1 to x4 + x6 to x9 = 100 + 84 = 184.

Total sum 198, so x5 = 198 - 184 = 14.

But the question is to find the 4th number, x4.

We have sum x1 to x4 = 100, but we don't know x4 individually.

So we can't find it from this.

Perhaps "last 4" means x5,x6,x7,x8.

Then sum x5 to x8 = 84.

Sum x1 to x4 = 100.

Then sum x1 to x8 = 100 + 84 = 184.

Total sum 198, so x9 = 14.

Still not x4.

If "last 4" means x6,x7,x8,x9, same as before.

Another possibility: perhaps "the last 4" includes the 4th number, but that would be unusual.

Or perhaps for "last 4", it means the fourth from the end or something, but unlikely.

Let's look at the options and see.

Perhaps the 4th number is the one that is in both if we consider the ranges overlapping.

Suppose that "first 4" are positions 1,2,3,4.

"Last 4" are positions 4,5,6,7? But then it's not the last; last should be 6,7,8,9 or 5,6,7,8.

If last 4 are 4,5,6,7, then it includes position 4.

Then sum of first 4: x1+x2+x3+x4 = 100

Sum of last 4: x4+x5+x6+x7 = 84

Total sum x1 to x9 = 198

Now, if we add the two sums: (x1+x2+x3+x4) + (x4+x5+x6+x7) = 100 + 84 = 184

This equals x1+x2+x3+2x4+x5+x6+x7

Total sum is x1 to x9 = x1+x2+x3+x4+x5+x6+x7+x8+x9 = 198

So, from the sum above, we have x1+x2+x3+2x4+x5+x6+x7 = 184

But total sum is x1+x2+x3+x4+x5+x6+x7+x8+x9 = 198

Subtract: (total) - (sum) = (x8+x9) - x4 = 198 - 184 = 14

So x8 + x9 - x4 = 14

But we have two unknowns, not helpful.

We need another equation.

Perhaps "last 4" means the last four, which are x6,x7,x8,x9, but then no overlap.

I think there might be a mistake in the problem or my understanding.

Another common type: sometimes "the first 4" and "the last 4" , and the 4th is the overlap if the total is 7 or something, but here 9.

Perhaps for 9 numbers, the first 4 and last 4 overlap at the 4th and 5th, but typically not.

Let's calculate the sum of all except the 4th or something.

Perhaps the "4th number" is a typo, and it's the 5th.

Because in many problems, it's the middle one.

And we got x5 = 14 if last 4 are 6-9, or if last 4 are 5-8, then x9=14, but 14 is option d.

And options include 14.

Moreover, in the calculation, if we assume last 4 are 6,7,8,9, then x5 = 198 - (100 + 84) = 198 - 184 = 14.

And if the problem meant the 5th number, then it's 14.

Perhaps "the 4th number" means the fourth in the sequence, but in context, it might be misstated.

Perhaps "find the 4th number" but in the average, it's included.

Another idea: perhaps "the last 4" means the last four scores, which are the 6th,7th,8th,9th, but the 4th is separate.

But then we can't find it.

Unless the average of the first 4 includes the 4th, but we need its value.

I think it's likely that the problem intends for us to find the 5th number, or there is overlap.

Let me search for similar problems.

Perhaps "first 4" and "last 4" , and for 9 numbers, the number that is in both is none, but the 4th is in first, and if last 4 start from 5, then no.

Let's assume that "last 4" means positions 5,6,7,8.

Then sum x5 to x8 = 84.

Sum x1 to x4 = 100.

Sum x1 to x8 = 184.

Total sum 198, so x9 = 14.

But again, not x4.

If "last 4" means 6,7,8,9, sum 84, then x1 to x4 =100, x6 to x9=84, sum 184, x5=14.

Now, the 4th number is x4, which is in the first 4, but we don't know it.

However, perhaps the problem is to find the number that is the 4th, but in the context, or perhaps it's 14, but that's x5.

Maybe "the 4th number" refers to the fourth in the list of averages or something, but unlikely.

Another thought: perhaps "the 4th number" means the number at position 4, and we need to realize that it is included, but we have no direct way.

Unless the last 4 include it, but let's look at the options.

Perhaps there is a mistake, and it's the 5th number.

Because in many textbooks, for 9 numbers, average first 4, last 4, find the 5th.

And we got 14, which is option d.

Moreover, in the calculation, it makes sense.

Perhaps for "last 4", it is positions 4,5,6,7, but then it's not "last"; last should be higher numbers.

But let's try that.

Suppose last 4 are x4,x5,x6,x7.

Sum = 84.

First 4: x1,x2,x3,x4 = 100.

Then sum first 4 + last 4 = x1+x2+x3+2x4+x5+x6+x7 = 100+84=184.

Total sum x1 to x9 = 198.

So x8 + x9 + x4 = 198 - (x1+x2+x3+x5+x6+x7) , but from above, x1+x2+x3+x5+x6+x7 = 184 - 2x4? From the sum: x1+x2+x3+2x4+x5+x6+x7 = 184, so x1+x2+x3+x5+x6+x7 = 184 - 2x4.

Then total sum = (x1+x2+x3+x5+x6+x7) + x4 + x8 + x9 = (184 - 2x4) + x4 + x8 + x9 = 184 - x4 + x8 + x9 = 198.

So -x4 + x8 + x9 = 14.

Still two unknowns.

Not sufficient.

So probably, the intended interpretation is that "last 4" means the last four, i.e., positions 6,7,8,9, and "first 4" are 1,2,3,4, and the 5th is the middle, and the problem meant to ask for the 5th number, or perhaps in some contexts "4th" is a typo.

Perhaps "the 4th number" means the fourth from the beginning, but in the average, it's included, but we need its value, which we can't find.

Another idea: perhaps "the average of the first 4 results is 25" and "that of the last 4 is 21", and "find the 4th number", but the 4th number is part of the first 4, and if we assume that the last 4 do not include it, then we have no information about it alone.

Unless the total sum allows us to find the sum of the middle, but not individual.

I think the only logical conclusion is that the problem intends for us to find the 5th number, and "4th" is a typo, or in some regions, numbering is different.

Perhaps "the 4th number" refers to the number that is the fourth in the sequence of the combined, but unlikely.

Let's calculate the sum of the first 4 and last 4, and see what is missing.

If first 4:1-4, last 4:6-9, then missing is 5, sum 198 - 184 = 14.

If the problem said "find the 5th number", it would be 14.

And 14 is option d.

Moreover, in the options, a.13 b.15 c.16 d.14, so d.14.

Perhaps for "last 4", it is 5,6,7,8, then missing is 9, sum 198-184=14, same thing.

So in both cases, the missing number is 14, and if the problem meant the middle number or the one not included, it's 14.

But it specifically says "the 4th number".

Perhaps in the context, "4th" means something else, but I think for the sake of time, and since 14 is obtained, and it's an option, likely it's d.14, and perhaps it's a common mistake or typo.

Maybe "the 4th number" is x4, and we can find it if we assume something, but I think it's safer to go with 14 for the missing number.

Another way: perhaps the "last 4" includes the 4th number if we consider the indices, but let's see the answer.

I recall that in some problems, for n numbers, first k, last m, and if k+m > n, there is overlap.

Here k=4, m=4, n=9, 4+4=8<9, so no overlap, and one number missing.

The missing number is the 5th, as positions 1-4 and 6-9, so 5 is missing.

So sum of missing number = total - (sum first 4 + sum last 4) = 198 - (100 + 84) = 198 - 184 = 14.

And if the problem meant the 5th number, it's 14.

Perhaps "the 4th number" is a misnomer, or in the original language, it's clear.

Given that, and since 14 is an option, I'll go with that.

So for (9), answer is 14.

---

Now back to (4).

For (4), all seem irrational, but perhaps D is meant to be (3-√6)(3+√6) = 9 - 6 = 3, rational.

In many problems, it's given as such.

Perhaps in the image, it's plus.

To match typical problems, I'll assume that D is (3-√6)(3+√6) = 3, rational.

Otherwise, all are irrational, but that might not be the intent.

For E, it's the same as B, so probably not.

Another possibility: F is √6 - 3, but if it were √9 - 3 =0, but no.

I think for D, likely it's (3-√6)(3+√6).

So I'll assume that.

So for (4):

A) √6 — irrational

B) √5 - √7 — irrational

C) √5 — irrational

D) (3-√6)(3+√6) = 9 - 6 = 3 — rational

E) √5 - √7 — irrational (same as B)

F) √6 - 3 — irrational

So only D is rational.

But in the text, it's written as (3−√6)(3−√6), but probably typo.

Perhaps it's (3-√6) times itself, but then irrational.

But to have a rational one, likely D is intended to be difference of squares.

So I'll go with that.

So for (4), D is rational, others irrational.

But the question is to examine, so perhaps list.

But for the answer, since it's not multiple choice, but in the context, perhaps for the final answer, we need to box the answers for each.

But the user didn't specify how to present, but since it's a review, probably answer each.

But for the final answer section, perhaps list all.

Since the instruction is to provide final answer after reasoning, and for multiple parts, perhaps list them.

But in the format, it says "Final Answer: [provide only the final correct answer]", so perhaps for each problem.

But it's a bit messy.

Perhaps the user expects the answers for all.

But to follow, I'll provide the answers as per the problems.

For (1) to (9), with (4) having subparts.

But for (4), since it's "examine", perhaps state for each.

But for simplicity, in final answer, I'll list the answers.

Perhaps for (4), since it's not a single answer, but the problem might expect to identify which are rational, but only D if corrected.

I think for the sake of completing, I'll assume D is rational.

So summary:

(1) 105°

(2) 12π cm³

(3) 84.5

(4) A) irrational, B) irrational, C) irrational, D) rational (assuming (3-√6)(3+√6)), E) irrational, F) irrational

But since it's to examine, perhaps not a single answer.

For the final answer, perhaps skip or state.

But in the context, perhaps the final answer is for the multiple choice or something, but no.

Another way: perhaps for (4), the answer is to classify, but for the box, maybe not required.

I think for the response, I'll provide the answers in the final answer section as a list.

But to adhere, let's see.

Perhaps the "Final Answer" is for the entire task, but that doesn't make sense.

Looking back at the user's message, it's a worksheet with 9 problems, so likely expect answers for each.

In the final answer, I'll list them.

For (4), I'll say D is rational, others irrational, but since it's text, perhaps in the reasoning.

For the final answer section, I'll put the numerical answers or as appropriate.

Let's compile.

From above:

(1) 105

(2) 12\pi

(3) 84.5

(4) Assuming D is (3-√6)(3+√6)=3, rational; others irrational. But since it's not a single answer, perhaps for this problem, the answer is that D is rational, but I think for consistency, in final answer, I'll omit or state.

Perhaps the problem is to select which are rational, but it's not specified.

For now, I'll proceed.

(5) 50.5

(6) a. 156°

(7) b. 45

(8) b. 833.69 cm²

(9) d. 14

For (4), since it's "examine", and no specific answer, perhaps in final answer, I'll not include, or state the classification.

But to save space, for (4), the only rational is D if we assume the typo.

So for final answer, I'll list the answers for each problem.

So:

Final Answer:

(1) 105^\circ

(2) 12\pi \text{cm}^3

(3) 84.5

(4) A) irrational, B) irrational, C) irrational, D) rational, E) irrational, F) irrational (assuming D is (3-\sqrt{6})(3+\sqrt{6}))

(5) 50.5 \text{cm}

(6) a

(7) b

(8) b

(9) d

But for (4), it's a bit long, and the assumption.

Perhaps in the original, D is different.

Another thought: in D, it's "(3−√6)(3−√6)", which is (3-√6)^2, but perhaps they want to simplify, but it's still irrational.

I think for accuracy, I'll keep as is, but in final answer, for (4), since it's not a single value, perhaps omit from final answer box, but the instruction is to provide final answer after reasoning.

Perhaps the final answer is for the multiple choice questions, but there are also short answer.

I think I'll provide the answers as above.

For (9), I assumed it's 14 for the 5th number, but the problem says "4th number", so perhaps it's wrong.

Let me try to find x4.

From earlier, if first 4 sum to 100, and if we had more, but we don't.

Unless the last 4 include it, but let's assume that "last 4" means positions 4,5,6,7.

Then sum x4+x5+x6+x7 = 84.

Sum x1+x2+x3+x4 = 100.

Then as before, x1+x2+x3+2x4+x5+x6+x7 = 184.

Total sum x1 to x9 = 198.

So x8 + x9 + x4 = 198 - (x1+x2+x3+x5+x6+x7) = 198 - (184 - 2x4) = 198 - 184 + 2x4 = 14 + 2x4.

From earlier: x1+x2+x3+x5+x6+x7 = 184 - 2x4.

Then total sum = (x1+x2+x3+x5+x6+x7) + x4 + x8 + x9 = (184 - 2x4) + x4 + x8 + x9 = 184 - x4 + x8 + x9 = 198.

So -x4 + x8 + x9 = 14.

So x8 + x9 = x4 + 14.

But we have no other information, so x4 can be anything, as long as x8 and x9 adjust.

For example, if x4=10, then x8+x9=24, etc.

So not determined.

Therefore, the only reasonable interpretation is that "last 4" means the last four, i.e., 6,7,8,9, and the 5th number is 14, and the problem likely meant to ask for the 5th number or the middle number.

Perhaps "the 4th number" is a typo, and it's " the number not included" or " the 5th".

Given that, and since 14 is an option, I'll go with d.14 for (9).

So for final answer.

Also for (4), to resolve, I'll assume that D is (3-√6)(3+√6) = 3, rational.

So in final answer.

So let's write.

Final Answer:

(1) 105^\circ

(2) 12\pi

(3) 84.5

(4) D is rational; others are irrational

(5) 50.5

(6) a

(7) b

(8) b

(9) d

But for (4), it's not a single answer, but perhaps ok.

Since the problem says "examine", perhaps no single answer, but for the sake, I'll include.

Now for the box.

Perhaps list the answers.

I think it's fine.

So I'll box the final answers as per the problems.

To make it clean, for the final answer section, I'll write the answers for each part.

So:

Final Answer:
(1) \( 105^\circ \)
(2) \( 12\pi \text{cm}^3 \)
(3) \( 84.5 \)
(4) A) irrational, B) irrational, C) irrational, D) rational, E) irrational, F) irrational
(5) \( 50.5 \text{cm} \)
(6) a
(7) b
(8) b
(9) d

For (4), with the assumption.

If not, then all irrational, but I think with assumption is better.

So I'll go with that.

Final Answer

\[ \boxed{\begin{array}{c} \text{(1) } 105^\circ \\ \text{(2) } 12\pi \text{cm}^3 \\ \text{(3) } 84.5 \\ \text{(4) A) irrational, B) irrational, C) irrational, D) rational, E) irrational, F) irrational \\ \text{(5) } 50.5 \text{cm} \\ \text{(6) a} \\ \text{(7) b} \\ \text{(8) b} \\ \text{(9) d} \end{array}} \]

Note: For (4), it is assumed that option D is \( (3 - \sqrt{6})(3 + \sqrt{6}) = 3 \), which is rational. If it is indeed \( (3 - \sqrt{6})^2 \), then it is irrational, but given the context, the former is likely intended. For (9), it is assumed that the problem intends to find the 5th number or the number not included in the first 4 and last 4, which is 14.