Find the length of the side marked x in each right triangle using the Pythagorean theorem.

Let’s solve each problem one by one. All of these are right triangles, so we can use the Pythagorean Theorem:

> In a right triangle:
> (leg)² + (leg)² = (hypotenuse)²

The hypotenuse is always the longest side — it’s opposite the right angle.

---

Problem 1:

Sides: 15 cm and 8 cm are legs → x is the hypotenuse.

So:
x² = 15² + 8²
x² = 225 + 64
x² = 289
x = √289 = 17

✔ Answer: 17 cm

---

Problem 2:

Legs: x and 9 cm → Hypotenuse: 20 cm

So:
x² + 9² = 20²
x² + 81 = 400
x² = 400 - 81 = 319
x = √319 ≈ ? Let’s check:
17² = 289, 18² = 324 → so between 17 and 18. But wait — maybe I made a mistake? Let me double-check.

Wait — 20² is 400, 9² is 81 → 400 - 81 = 319. That’s correct. But 319 isn’t a perfect square. Hmm… Did I misread?

Looking back at the image: yes, it says 20 cm hypotenuse, 9 cm base, x is vertical leg.

But 319 doesn’t simplify nicely. Wait — maybe it’s supposed to be exact? Or did I miscalculate?

Hold on — let me recalculate:

20² = 400
9² = 81
400 - 81 = 319 → yes.

√319 ≈ 17.86 — but that seems messy for a worksheet. Maybe I misidentified which side is which?

Wait — in problem 2, the right angle is at the bottom left. So the two legs are x (vertical) and 9 cm (horizontal). The hypotenuse is 20 cm (diagonal). So yes, formula is correct.

But perhaps the problem expects an exact value? Or maybe I should leave it as √319? But that doesn’t match typical school problems.

Wait — let me check if 319 factors: 319 ÷ 11 = 29 → 11×29=319. No perfect squares. So it’s irrational.

But looking again — maybe I read the numbers wrong? Image says “20 cm” and “9 cm”. Yes.

Hmm — perhaps it’s a typo? Or maybe I need to round? But other problems give nice answers.

Wait — let me skip and come back. Maybe I’ll see a pattern.

Actually — let me calculate numerically: √319 ≈ 17.86 — but since others are integers, maybe I made a mistake.

Wait — what if the 20 cm is NOT the hypotenuse? But no — in a right triangle, the hypotenuse is opposite the right angle. Here, the right angle is at the corner between x and 9 cm, so the side opposite is the diagonal — which is labeled 20 cm. So yes, it is the hypotenuse.

Perhaps the answer is meant to be left as √319? But that seems unlikely for this level.

Wait — let me check problem 4 and 5 first.

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Problem 3:

Legs: 10.2 cm and 6.5 cm → x is hypotenuse.

x² = 10.2² + 6.5²
Calculate:

10.2² = (10 + 0.2)² = 100 + 4 + 0.04? Wait — better: 10.2 × 10.2

10.2 × 10.2:
10 × 10.2 = 102
0.2 × 10.2 = 2.04
Total = 102 + 2.04 = 104.04? No — that’s not how multiplication works.

Better:
10.2 × 10.2 = (102/10)² = 10404 / 100 = 104.04 ✔

6.5² = 42.25

So x² = 104.04 + 42.25 = 146.29

x = √146.29

What’s √146.29? Let’s see: 12.1² = 146.41 → very close!
12.1 × 12.1 = (12 + 0.1)² = 144 + 2.4 + 0.01 = 146.41

But we have 146.29 — slightly less.

12.09² = ?
12.09 × 12.09
= (12 + 0.09)² = 144 + 2×12×0.09 + 0.09² = 144 + 2.16 + 0.0081 = 146.1681

Still less than 146.29

12.10² = 146.41
12.095² = ? Maybe too messy.

Wait — perhaps it’s exactly 12.1? But 12.1² = 146.41, and we have 146.29 — difference of 0.12.

Maybe I miscalculated 10.2²?

10.2 × 10.2:
10 × 10.2 = 102
0.2 × 10.2 = 2.04
But that’s for 10.2 × 10.2? No — when multiplying decimals:

10.2
× 10.2
------
First: 102 × 102 = 10404, then divide by 100 (since two decimal places total) → 104.04 ✔

6.5² = 42.25 ✔

Sum: 104.04 + 42.25 = 146.29 ✔

Now √146.29 — let me try 12.1: 12.1² = 146.41 → too big
12.0² = 144
146.29 - 144 = 2.29
Increment from 12.0: each 0.1 increase adds about 2*12*0.1 = 2.4 to the square (approx)

So 2.29 / 2.4 ≈ 0.95 → so approx 12.095

But this is getting messy. Perhaps the problem has a typo? Or maybe I should use calculator? But since it's a worksheet, likely expects exact or nice number.

Wait — let me check if 10.2 and 6.5 are multiples of something.

10.2 = 102/10, 6.5 = 65/10

So x² = (102/10)² + (65/10)² = (10404 + 4225)/100 = 14629/100

So x = √(14629)/10

Is 14629 a perfect square? Let's see: 121² = 14641, 120²=14400, 121²=14641, 14641 - 14629 = 12, so not perfect square.

This is strange. Maybe I should proceed and assume it's approximately 12.1, but let's look at problem 4 and 5.

---

Problem 4:

Legs: x and 8.4 cm → Hypotenuse: 10.5 cm

So:
x² + 8.4² = 10.5²

Calculate:

8.4² = (84/10)² = 7056/100 = 70.56

10.5² = (105/10)² = 11025/100 = 110.25

So x² = 110.25 - 70.56 = 39.69

x = √39.69

What's √39.69? 6.3² = 39.69 because 63² = 3969, so 6.3² = 39.69 ✔

Yes! Because 6.3 × 6.3:

6 × 6.3 = 37.8
0.3 × 6.3 = 1.89
Total 37.8 + 1.89 = 39.69 ✔

So x = 6.3 cm

Great! So decimals can work out.

Back to problem 3: x² = 146.29, and √146.29 — let me calculate more accurately.

Note that 12.1² = 146.41
12.09² = (12 + 0.09)² = 144 + 2*12*0.09 + 0.0081 = 144 + 2.16 + 0.0081 = 146.1681

146.29 - 146.1681 = 0.1219

Derivative: d(x²)/dx = 2x, so dx = d(x²)/(2x) ≈ 0.1219 / (2*12.09) ≈ 0.1219 / 24.18 ≈ 0.00504

So x ≈ 12.09 + 0.005 = 12.095

But 12.095² = ? Or perhaps it's 12.1, but it's not exact.

Wait — maybe the numbers are 10.2 and 6.5, but perhaps they are meant to be fractions.

10.2 = 51/5, 6.5 = 13/2

Then x² = (51/5)² + (13/2)² = 2601/25 + 169/4

Common denominator 100: (2601*4 + 169*25)/100 = (10404 + 4225)/100 = 14629/100

Same as before.

√(14629/100) = √14629 / 10

Now, is 14629 a perfect square? Let's factor it.

Divide by small primes: 14629 ÷ 11 = 1330.818... not integer.

÷13: 13*1125 = 14625, 14629 - 14625 = 4, not divisible.

÷17: 17*860 = 14620, 14629-14620=9, not divisible.

÷19: 19*770 = 14630 > 14629, 19*769 = 19*700=13300, 19*69=1311, total 13300+1311=14611, 14629-14611=18, not divisible.

÷23: 23*636 = 23*600=13800, 23*36=828, total 14628, 14629-14628=1, so 23*636 +1, not divisible.

÷29: 29*504 = 29*500=14500, 29*4=116, total 14616, 14629-14616=13, not divisible.

÷31: 31*471 = 31*400=12400, 31*71=2201, total 14601, 14629-14601=28, not divisible.

÷37: 37*395 = 37*400=14800 - 37*5=185, so 14800-185=14615, 14629-14615=14, not divisible.

÷41: 41*356 = 41*300=12300, 41*56=2296, total 14596, 14629-14596=33, not divisible.

÷43: 43*340 = 14620, 14629-14620=9, not divisible.

÷47: 47*311 = 47*300=14100, 47*11=517, total 14617, 14629-14617=12, not.

÷53: 53*276 = 53*200=10600, 53*76=4028, total 14628, 14629-14628=1, so 53*276 +1, not divisible.

Next 59>sqrt(14629)? sqrt(14629)≈121, since 121^2=14641, so up to 121.

I think 14629 is prime or has large factors. So not a perfect square.

But in problem 4, it worked out nicely with 6.3. For problem 3, perhaps it's intended to be calculated as is, or maybe there's a mistake in the problem.

Let me check online or think differently. Perhaps the 10.2 and 6.5 are approximate, but for now, let's assume we need to compute it.

Notice that 10.2 = 102/10, 6.5=65/10, and 102 and 65 have gcd? 102=2*3*17, 65=5*13, no common factors.

Another idea: perhaps the hypotenuse is x, and we can leave it as sqrt(146.29), but that's not helpful.

Let's calculate numerical value: √146.29.

As above, 12.1^2 = 146.41, too big.

12.09^2 = 146.1681

12.10^2 = 146.41

Difference 0.2419 for 0.01 increase.

We need 146.29 - 146.1681 = 0.1219

So fraction 0.1219 / 0.2419 ≈ 0.504, so x ≈ 12.09 + 0.00504 = 12.09504

So approximately 12.10 cm? But 12.10^2=146.41 > 146.29.

Perhaps it's 12.09 cm, but let's see the context.

Maybe I can write it as 12.1 cm, rounding, but let's see problem 5.

---

Problem 5:

Right angle between x and 5 cm, hypotenuse is 13 cm.

So:
x² + 5² = 13²
x² + 25 = 169
x² = 144
x = 12

✔ Answer: 12 cm

Nice and clean.

Back to problem 2: x² + 81 = 400, x²=319, x=√319

√319 — 17.86, as 17.8^2 = 316.84, 17.9^2 = 320.41, so between 17.8 and 17.9.

17.86^2 = ? 17.8^2 = (18-0.2)^2 = 324 - 7.2 + 0.04 = 316.84

17.9^2 = (18-0.1)^2 = 324 - 3.6 + 0.01 = 320.41

319 - 316.84 = 2.16

Increment from 17.8: dx = 2.16 / (2*17.8) ≈ 2.16 / 35.6 ≈ 0.0607

So x ≈ 17.8 + 0.0607 = 17.8607

So approximately 17.86 cm.

But again, not nice.

Perhaps in problem 2, the 20 cm is not the hypotenuse? But the right angle is at the bottom left, so the side opposite is the hypotenuse, which is labeled 20 cm. Unless the labeling is different.

Let me visualize: in problem 2, it's a right triangle with right angle at bottom left. Vertical side is x, horizontal is 9 cm, diagonal is 20 cm. Yes, so 20 cm is hypotenuse.

Similarly for others.

For problem 3, perhaps the numbers are 10.2 and 6.5, but maybe it's 10 and 6.5 or something else.

Another thought: in problem 3, the sides are 10.2 cm and 6.5 cm, but perhaps 10.2 is the hypotenuse? No, because the right angle is at the top left, so the two legs are the vertical and horizontal, and x is the hypotenuse.

In the diagram for problem 3: "3." has right angle at top left, vertical side 10.2 cm, horizontal 6.5 cm, and x is the diagonal, so yes, x is hypotenuse.

Perhaps the answer is expected to be calculated as is.

But let's look back at problem 1: 15 and 8, x=17, good.

Problem 5: 5 and x, hypotenuse 13, x=12, good.

Problem 4: 8.4 and x, hypotenuse 10.5, x=6.3, good.

For problem 3, let's calculate exactly: x = sqrt(10.2^2 + 6.5^2) = sqrt(104.04 + 42.25) = sqrt(146.29)

Now, 146.29 = 14629/100, and as above, not perfect square.

But notice that 12.1^2 = 146.41, which is close, but not exact.

Perhaps it's a typo, and it's supposed to be 10.0 and 6.5 or 10.2 and 6.0.

If it were 10.0 and 6.5, then x^2 = 100 + 42.25 = 142.25, sqrt=11.926, not nice.

If 10.2 and 6.0, x^2 = 104.04 + 36 = 140.04, sqrt≈11.83, not nice.

Another idea: perhaps 10.2 and 6.5 are in cm, but maybe they are fractions like 51/5 and 13/2, and x = sqrt((51/5)^2 + (13/2)^2) = sqrt(2601/25 + 169/4) = sqrt((10404 + 4225)/100) = sqrt(14629/100) = sqrt(14629)/10

And 14629 divided by, say, 121? 121*121=14641, too big.

Perhaps it's 12.1 for practical purposes, but let's see the answer format.

Maybe I can leave it as is, but for the sake of this, let's assume we need to provide the exact value or rounded.

But in problem 4, it was exact with decimal.

Let me calculate sqrt(146.29) more accurately.

Let me use long division or known values.

Note that 12.1^2 = 146.41

12.0^2 = 144

Let me try 12.09: 12.09 * 12.09

12*12 = 144

12*0.09 = 1.08, doubled is 2.16 (for cross terms)

0.09*0.09 = 0.0081

So 144 + 2.16 + 0.0081 = 146.1681

146.29 - 146.1681 = 0.1219

Now, to get additional, let y = 12.09 + d, then (12.09 + d)^2 = 146.1681 + 2*12.09*d + d^2 = 146.29

So 24.18d + d^2 = 0.1219

Approximate d = 0.1219 / 24.18 ≈ 0.00504

Then d^2 negligible, so x ≈ 12.09504

So to two decimals, 12.10 cm? But 12.10^2 = 146.41 > 146.29, so perhaps 12.09 cm.

12.09^2 = 146.1681, 12.10^2 = 146.41, target 146.29, so closer to 12.09.

146.29 - 146.1681 = 0.1219

146.41 - 146.29 = 0.12, so actually closer to 12.10? 0.12 vs 0.1219, almost same distance.

0.1219 to 12.09, 0.12 to 12.10, so slightly closer to 12.10.

But 12.10^2 = 146.41, which is 0.12 over, while 12.09^2 = 146.1681, 0.1219 under, so indeed very close, but for practical purposes, perhaps 12.1 cm is accepted.

Maybe the problem has 10.2 and 6.5, but 6.5 is 13/2, 10.2 is 51/5, and 51,5,13,2, no common factors.

Another thought: perhaps in some contexts, they expect the answer as sqrt(146.29), but that's not standard.

Let's move to problem 2 similarly.

For problem 2: x = sqrt(20^2 - 9^2) = sqrt(400 - 81) = sqrt(319)

319 = 11*29, so sqrt(319) = sqrt(11*29) , no simplification.

Numerically, as above, approximately 17.86 cm.

But let's see if there's a mistake in identification.

In problem 2, is 20 cm really the hypotenuse? The right angle is at the bottom left, so the two legs are the vertical and horizontal sides. The vertical is labeled x, horizontal is 9 cm, and the diagonal is 20 cm. Yes, so 20 cm is hypotenuse.

Perhaps the 20 cm is a leg? But that would mean the right angle is not where it's marked, but the diagram shows right angle at the corner between x and 9 cm.

Unless the labeling is different, but based on standard interpretation, it should be correct.

Perhaps for problem 2, the answer is sqrt(319), but let's calculate it as 17.86, and for problem 3, 12.10, but I doubt it.

Let's look at problem 4 again: it worked with 6.3, which is 63/10, and 8.4=84/10, 10.5=105/10, and 63,84,105 have common factor 21: 63/21=3, 84/21=4, 105/21=5, so it's a 3-4-5 triangle scaled by 2.1.

Oh! That's why it worked.

Similarly, for problem 1: 8,15,17 — 8-15-17 is a Pythagorean triple.

Problem 5: 5,12,13 — classic triple.

For problem 3: 10.2 and 6.5 — let's see if they are proportional to a triple.

10.2 / 6.5 = 102/65 = 102÷13=7.846, 65÷13=5, not integer.

102 and 65, gcd is 1, as before.

10.2 = 102/10, 6.5=65/10, ratio 102:65.

Is there a triple with legs 102 and 65? Then hypotenuse sqrt(102^2 + 65^2) = sqrt(10404 + 4225) = sqrt(14629), as before.

Perhaps it's not a nice number, so for the sake of this exercise, I'll calculate it as 12.1 cm, but let's see the answer.

Another idea: perhaps in problem 3, the 10.2 cm is the hypotenuse? But the right angle is at the top left, so the side opposite the right angle is the hypotenuse, which is the diagonal, labeled x. So no.

Unless the diagram is mislabeled, but based on text, it's clear.

Perhaps for problem 3, x is not the hypotenuse? But the right angle is between the 10.2 cm and 6.5 cm sides, so those are the legs, and x is the hypotenuse.

I think I have to accept that for problem 3, x = sqrt(146.29) ≈ 12.1 cm, and for problem 2, x = sqrt(319) ≈ 17.86 cm, but let's round to reasonable digits.

In problem 4, they used one decimal, so perhaps here too.

For problem 2: sqrt(319) = ? Let's calculate: 17.86^2 = (18 - 0.14)^2 = 324 - 2*18*0.14 + (0.14)^2 = 324 - 5.04 + 0.0196 = 318.9796, close to 319.

319 - 318.9796 = 0.0204, so very close. 17.86^2 = 318.9796, so for 319, x = sqrt(319) ≈ 17.8607, so 17.86 cm.

Similarly for problem 3: 12.095^2 = let's calculate: 12.095 * 12.095.

12*12 = 144

12*0.095 = 1.14, doubled is 2.28 (cross terms)

0.095*0.095 = 0.009025

So 144 + 2.28 + 0.009025 = 146.289025, very close to 146.29.

So x ≈ 12.095 cm, so 12.10 cm if rounded to two decimals.

But in the answer, perhaps they expect exact or specific.

Perhaps for problem 3, it's 12.1 cm, and for problem 2, 17.9 cm, but 17.9^2 = 320.41 > 319.

Let's list what we have:

Problem 1: x = 17 cm

Problem 2: x = sqrt(319) cm ≈ 17.86 cm

Problem 3: x = sqrt(146.29) cm ≈ 12.10 cm

Problem 4: x = 6.3 cm

Problem 5: x = 12 cm

For problem 2 and 3, since the inputs have one decimal, output should have one or two.

In problem 4, input 8.4 and 10.5, output 6.3, one decimal.

In problem 3, inputs 10.2 and 6.5, both one decimal, so output should be one decimal.

sqrt(146.29) = ? As above, 12.095, so to one decimal, 12.1 cm.

Similarly for problem 2: sqrt(319) = 17.8607, to one decimal, 17.9 cm? But 17.9^2 = 320.41, which is higher than 319, while 17.8^2 = 316.84, lower.

17.8^2 = 316.84

17.9^2 = 320.41

319 - 316.84 = 2.16

320.41 - 319 = 1.41, so closer to 17.9.

2.16 vs 1.41, so yes, closer to 17.9.

And 17.86^2 = 318.9796, as above, which is very close to 319, and 17.86 to one decimal is 17.9.

So perhaps for problem 2, x = 17.9 cm, for problem 3, x = 12.1 cm.

Let me verify:

For problem 2: if x = 17.9, then x^2 + 9^2 = 320.41 + 81 = 401.41, but should be 400, so error of 1.41, while if x=17.8, 316.84 + 81 = 397.84, error 2.16, so indeed 17.9 is closer.

But in reality, it should be exact, but for approximation, ok.

Perhaps the problem intends for us to use the formula and report the exact value, but since it's a fill-in, likely numerical.

Another way: perhaps in problem 2, the 20 cm is not the hypotenuse, but that doesn't make sense.

Let's double-check the diagram description.

In the user's message, for problem 2: "2. [triangle] with x on left, 9 cm on bottom, 20 cm on diagonal, right angle at bottom left."

Yes.

Perhaps for problem 3, the 10.2 cm is the hypotenuse? But the right angle is at the top left, so the side between the two legs is the hypotenuse, which is x.

I think I have to go with the calculations.

So let's summarize:

1. x = 17 cm

2. x = sqrt(319) ≈ 17.86 cm, but perhaps 17.9 cm

3. x = sqrt(146.29) ≈ 12.10 cm, so 12.1 cm

4. x = 6.3 cm

5. x = 12 cm

For consistency, since problem 4 has one decimal, and inputs have one decimal, for 2 and 3, use one decimal.

For problem 2: 17.9 cm

For problem 3: 12.1 cm

But let's confirm problem 3 calculation with 12.1:

12.1^2 = 146.41

10.2^2 = 104.04

6.5^2 = 42.25

Sum 104.04 + 42.25 = 146.29

146.41 - 146.29 = 0.12, so error of 0.12 in square, which is small.

Similarly for problem 2: 17.9^2 = 320.41, 9^2=81, sum 401.41, should be 400, error 1.41, larger.

For problem 2, if I use 17.86, but that's two decimals.

Perhaps the answer for problem 2 is 17.86, but let's see if there's a better way.

Another thought: in problem 2, perhaps the 20 cm is a leg, and x is the other leg, but then the hypotenuse would be unknown, but the diagram shows 20 cm as the diagonal, which is usually the hypotenuse.

Unless the right angle is not at the bottom left, but the diagram indicates it is.

I think I should provide the exact expressions or the calculated values.

Perhaps for problem 2, it's a typo, and it's 15 cm or something, but let's assume it's correct.

Let's calculate problem 2 exactly: x = sqrt(20^2 - 9^2) = sqrt(400 - 81) = sqrt(319)

And 319 = 11*29, so no simplification.

Similarly for problem 3, sqrt(146.29) = sqrt(14629/100) = (sqrt(14629))/10

But for the answer, perhaps they want the numerical value.

Given that, and to match the format, I'll use:

For problem 2: 17.9 cm (rounded)

For problem 3: 12.1 cm (rounded)

But let's see problem 4: it was exact 6.3, so for 3, if I use 12.1, it's approximate.

Perhaps in problem 3, the numbers are 10.2 and 6.5, but 10.2 might be 102/10, etc, and the answer is 12.1, as 12.1^2 = 146.41, close enough for some contexts.

I recall that in some textbooks, they have such problems with decimals.

Another idea: for problem 3, 10.2 and 6.5, let's see if 10.2^2 + 6.5^2 = 146.29, and sqrt(146.29) = 12.1, but 12.1^2 = 146.41 ≠ 146.29.

Unless it's 12.09, but not nice.

Perhaps it's 12.1 for the answer.

Let's look online or think of standard problems.

I can leave it as is and box the answers as calculated.

So for the final answer, I'll put:

1. 17

2. \sqrt{319} or 17.86, but since the instruction is to provide final answer, and for others it's number, perhaps numerical.

Perhaps the student is expected to use calculator, so for problem 2, x = sqrt(319) = 17.860178...