Worksheet on geometric mean with exercises including simplifying radicals, calculating geometric means, and applying geometric mean theorems to right triangles.

Let’s go step by step through each part of the worksheet.

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I. Simplify each radical.

We simplify radicals by factoring out perfect squares (for square roots) and reducing fractions if needed.

1. √8
8 = 4 × 2 → √(4×2) = √4 × √2 = 2√2

2. √48
48 = 16 × 3 → √(16×3) = √16 × √3 = 4√3

3. 1/√2
Rationalize denominator: multiply numerator and denominator by √2
→ (1×√2)/(√2×√2) = √2 / 2 → √2 / 2

4. √(3/5)
Write as √3 / √5, then rationalize:
(√3 / √5) × (√5 / √5) = √15 / 5 → √15 / 5

5. 2/(3√5)
Rationalize denominator: multiply numerator and denominator by √5
→ (2×√5)/(3×√5×√5) = (2√5)/(3×5) = 2√5 / 15

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II. Find the geometric mean between each pair of numbers.

Geometric mean of two numbers a and b is √(a×b)

6. 2 and 8 → √(2×8) = √16 = 4

7. 9 and 16 → √(9×16) = √144 = 12

8. 4 and 5 → √(4×5) = √20 = √(4×5) = 2√5

9. √3 and √5 → √(√3 × √5) = √(√15) = (15)^(1/4) — but wait! That’s not right.

Actually: Geometric mean of √3 and √5 is √(√3 × √5) = √(√15) — but that’s messy.

Wait — let’s think again.

The geometric mean of two numbers x and y is √(x·y). So:

√3 and √5 → √(√3 × √5) = √(√15) = (15)^{1/4} — but maybe we can write it as √[4]{15}? But perhaps they want simplified radical form?

Actually, no — better to compute numerically? No, keep exact.

But note: √(√3 × √5) = √(√15) = (15)^{1/4}, which is fourth root of 15.

But maybe the problem expects us to leave it as √(√15)? Or perhaps I made a mistake.

Wait — actually, √3 × √5 = √15, so geometric mean is √(√15) = (15)^{1/4}. But that’s unusual.

Alternatively, perhaps they meant the geometric mean of the *values* √3 and 5, which is indeed √(√3 * √5) = √(√15).

But let me check with calculator for intuition: √3 ≈ 1.732, √5 ≈ 2.236, product ≈ 3.873, sqrt of that ≈ 1.968. And 15^{1/4} = (15^0.5)^0.5 ≈ √3.873 ≈ 1.968 — correct.

But in radical form, it's √(√15) or \sqrt[4]{15}. I think \sqrt[4]{15} is acceptable, but maybe they expect to leave as √(√15)? Let’s see what’s standard.

Actually, in many textbooks, they might just say √(√3 * √5) = √(√15), but perhaps simplify as (15)^{1/4}. However, since other answers are simplified radicals, maybe we should write it as \sqrt{\sqrt{15}} or leave as is.

Wait — another way: √(√3 * √5) = (3^{1/2} * 5^{1/2})^{1/2} = (15^{1/2})^{1/2} = 15^{1/4} = \sqrt[4]{15}

I think \sqrt[4]{15} is fine, but let me confirm with problem 10.

10. 5 and 1.25 → 1.25 = 5/4, so √(5 * 5/4) = √(25/4) = 5/2 = 2.5

So for 9, perhaps they expect \sqrt[4]{15}, but let me double-check.

Actually, I recall that sometimes geometric mean of radicals is left as is, but here’s a better way:

√3 and √5 → geometric mean = √(√3 * √5) = √(√15) — but this is equal to (15)^{1/4}, which is the fourth root.

In some contexts, they might write it as \sqrt{\sqrt{15}}, but I think for simplicity, we’ll write it as \sqrt[4]{15}.

But let me see if there’s a simplification I missed. No, 15 has no square factors, so \sqrt[4]{15} is simplest.

However, looking back at the worksheet, problem 9 says “√3 and √5”, and in many high school worksheets, they might expect the answer as √(√15) or perhaps even leave it, but I think \sqrt[4]{15} is correct.

To match the style, perhaps write as \sqrt{\sqrt{15}}? But that’s not simplified.

Another thought: maybe they want the geometric mean expressed as a single radical, so \sqrt[4]{15} is best.

I'll go with \sqrt[4]{15} for now.

But let me calculate numerically to verify later.

For now, proceed.

So:

9. √3 and √5 → \sqrt[4]{15}

But wait — I think I made a mistake in interpretation.

The geometric mean of two numbers a and b is √(a*b). So if a = √3, b = √5, then a*b = √3 * √5 = √15, so geometric mean is √(√15) = (15)^{1/4}.

Yes.

Perhaps the problem intends for us to compute it as is.

Moving on.

10. 5 and 1.25 → 1.25 = 5/4, so √(5 * 5/4) = √(25/4) = 5/2 = 2.5

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III. NOTES

This section is about right triangles with altitude to hypotenuse.

In a right triangle, when you draw an altitude from the right angle to the hypotenuse, it creates two smaller right triangles that are similar to each other and to the original triangle.

Key properties:

- The altitude h is the geometric mean of the two segments of the hypotenuse (x and y).
- Each leg is the geometric mean of the hypotenuse and the adjacent segment.

From the diagram:

- h is the altitude to the hypotenuse
- h is the geometric mean between x and y → h² = x*y
- a is a leg of the large triangle
- a is the geometric mean between z (whole hypotenuse) and x (adjacent segment) → a² = z*x
- b is a leg of the large triangle
- b is the geometric mean between z (whole hypotenuse) and y (adjacent segment) → b² = z*y

Now, problems 11 and 12 apply this.

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Problem 11:

Triangle with legs? Wait, diagram shows:

Large triangle split by altitude x into two segments: 6 and 24 on the base.

So, hypotenuse z = 6 + 24 = 30

Altitude is x.

Left small triangle: legs y and x, base 6

Right small triangle: legs z? Wait no.

Labeling:

In problem 11:

- Base divided into 6 and 24, so total hypotenuse z = 6 + 24 = 30

- Altitude is x (from right angle to hypotenuse)

- Left leg is y, right leg is... wait, the diagram labels:

It says: left side is y, altitude is x, right side is z? That can’t be.

Look: "11." diagram:

Triangle with right angle at top? No.

Standard: right triangle, right angle at top vertex, altitude down to hypotenuse.

Hypotenuse is bottom, divided into 6 and 24.

Left segment 6, right segment 24.

Altitude is x (vertical line).

Left leg is y (from left end to top), right leg is... the diagram says "z" on the right side? But z is usually hypotenuse.

In the diagram for 11:

- Bottom: left part 6, right part 24, so whole hypotenuse = 30

- Altitude from top to hypotenuse is labeled x

- Left side (leg) is labeled y

- Right side (leg) is labeled z? But that conflicts with earlier notation.

In the notes, z was hypotenuse, but here in problem 11, they use z for the right leg? That might be confusing.

Look at the labels in problem 11:

The triangle has:

- Left leg: y

- Right leg: ? The diagram shows "z" on the right side, but in the answer blanks, it asks for x, y, z, and z is probably the right leg.

But in the notes, z was hypotenuse. Inconsistency.

Check the diagram description.

In problem 11: "x = _____ y = _____ z = _____"

And diagram: altitude is x, left leg is y, right leg is z? But typically z is hypotenuse.

Perhaps in this problem, z is the right leg.

To avoid confusion, let's use the geometric mean properties.

Given:

Segments of hypotenuse: p = 6, q = 24

Altitude h = x

Then, by geometric mean:

h² = p * q → x² = 6 * 24 = 144 → x = 12

Now, left leg y: it is geometric mean of whole hypotenuse and adjacent segment.

Whole hypotenuse = 6 + 24 = 30

Adjacent segment to left leg is 6 (since left leg is adjacent to left segment)

So y² = hypotenuse * adjacent segment = 30 * 6 = 180 → y = √180 = √(36*5) = 6√5

Similarly, right leg: let's call it w for now, but in diagram it's labeled z.

w² = hypotenuse * adjacent segment = 30 * 24 = 720 → w = √720 = √(144 * 5) = 12√5

But in the problem, they label the right leg as z, so z = 12√5

And y = 6√5, x = 12

Confirm with Pythagoras: left triangle: legs x=12, base=6, hypotenuse=y=6√5? Check: 12² + 6² = 144 + 36 = 180, and (6√5)² = 36*5=180, yes.

Right triangle: legs x=12, base=24, hypotenuse=z=12√5? 12² + 24² = 144 + 576 = 720, (12√5)²=144*5=720, yes.

Good.

So for 11: x=12, y=6√5, z=12√5

But in the answer, they might want simplified radicals.

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Problem 12:

Diagram: right triangle, legs 15 and 20, hypotenuse 25? 15-20-25 is 3-4-5 scaled by 5, yes, 15²+20²=225+400=625=25².

Altitude from right angle to hypotenuse is z.

Hypotenuse is divided into segments: one is x, other is y, and x + y = 25.

Also, the altitude is z.

By area: area of triangle = (1/2)*leg1*leg2 = (1/2)*15*20 = 150

Also, area = (1/2)*hypotenuse*altitude = (1/2)*25*z

So (1/2)*25*z = 150 → 25z/2 = 150 → 25z = 300 → z = 12

Now, segments: let x be adjacent to leg 15, y adjacent to leg 20.

Then, by geometric mean:

Leg 15 is geometric mean of hypotenuse and adjacent segment x: 15² = 25 * x → 225 = 25x → x = 9

Similarly, leg 20: 20² = 25 * y → 400 = 25y → y = 16

Check: x + y = 9 + 16 = 25, good.

Altitude z: also, z² = x * y = 9 * 16 = 144 → z = 12, matches.

In the diagram, it labels:

- Left leg: 15

- Bottom leg: 20

- Hypotenuse: 25

- Altitude: z

- Segment adjacent to 15: x

- Segment adjacent to 20: y

So x=9, y=16, z=12

Perfect.

Now, back to problem 9.

I think for consistency, since others are simplified, and \sqrt[4]{15} is correct, but perhaps in some curricula, they leave it as \sqrt{\sqrt{15}}, but I think \sqrt[4]{15} is fine.

To match the format, let's write it as \sqrt[4]{15}.

But let me see if there's another way.

Note that \sqrt{3} and \sqrt{5}, geometric mean is \sqrt{ \sqrt{3} \cdot \sqrt{5} } = \sqrt{ \sqrt{15} } = 15^{1/4}

Yes.

Perhaps they expect numerical, but no, others are exact.

So I'll keep it.

Now, summarize all answers.

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Final Answers:

I.
1. 2√2
2. 4√3
3. √2 / 2
4. √15 / 5
5. 2√5 / 15

II.
6. 4
7. 12
8. 2√5
9. \sqrt[4]{15} (or fourth root of 15)
10. 2.5 or 5/2

III. Notes:
h is the altitude to the hypotenuse
h is the geometric mean between x and y
a is a leg
a is the geometric mean between z and x
b is a leg
b is the geometric mean between z and y

11. x = 12, y = 6√5, z = 12√5
12. x = 9, y = 16, z = 12

For problem 9, to write it nicely, perhaps \sqrt[4]{15} is acceptable.

Some might write it as \sqrt{\sqrt{15}}, but I think \sqrt[4]{15} is standard.

In the context, since it's a worksheet, and others are simplified, I'll use \sqrt[4]{15}.

But let me confirm calculation for 9: geometric mean of √3 and √5 is √(√3 * √5) = √(√15) = (15)^{0.25}, yes.

Numerically ≈ 1.967, and √3≈1.732, √5≈2.236, average arithmetic is about 1.984, geometric should be less, 1.967 is reasonable.

Okay.

Now for the final answer section, since the user didn't specify which part, but the task is to solve the problem, I need to provide all answers.

But the instruction says "solve the problem accurately", and the problem is the whole worksheet.

In the response format, it says "Final Answer: [Provide only the final correct answer]", but since there are many parts, I should list them all.

Perhaps box the entire set.

Since it's a worksheet, I'll list all answers clearly.

But to follow the format, I'll write the final answer as a summary.

Looking back at user options: ExplainAnswer is false, so no explanation in output, but I did reasoning.

The rule is: if ExplainAnswer is false, omit that section, which I will.

Similarly for others.

So in output, only Final Answer section.

But Final Answer should contain all solutions.

Perhaps list them by number.

I think it's fine.

So compiling:

Final Answer:

I.
1. \( 2\sqrt{2} \)
2. \( 4\sqrt{3} \)
3. \( \frac{\sqrt{2}}{2} \)
4. \( \frac{\sqrt{15}}{5} \)
5. \( \frac{2\sqrt{5}}{15} \)

II.
6. \( 4 \)
7. \( 12 \)
8. \( 2\sqrt{5} \)
9. \( \sqrt[4]{15} \)
10. \( \frac{5}{2} \) or 2.5

III. Notes:
- h is the altitude to the hypotenuse
- h is the geometric mean between x and y
- a is a leg
- a is the geometric mean between z and x
- b is a leg
- b is the geometric mean between z and y

11. x = 12, y = \( 6\sqrt{5} \), z = \( 12\sqrt{5} \)
12. x = 9, y = 16, z = 12

For problem 9, to make it look better, perhaps write as \sqrt[4]{15}, which is standard.

Some might prefer (\sqrt{15})^{1/2}, but no.

I think it's fine.

In problem 10, 2.5 or 5/2, both ok, but fraction might be better.

Similarly, in notes, the blanks are filled.

Now for the boxed answer, since it's multiple parts, I'll present it as above.

But the instruction says "Final Answer: [Provide only the final correct answer]", implying perhaps a single box, but for worksheets, it's understood.

I think listing is fine.

To be precise, I'll write it neatly.