Question

the given lengths are two sides of a right triangle. all three side lengths of the triangle are integers and together form a pythagorean triple. find the length of the third side and tell whether it is a leg or the hypotenuse.

1. 15 and 36 \t\t\t\t 6. 70 and 250 \t\t\t\t 7. 45 and 51

1. 15 and 20 \t\t\t\t 9. 96 and 100 \t\t\t\t 10. 36 and 60

find the missing side length. leave answers as simplified radicals.

1. \t\t\t\t\t\t 12.

1. \t\t\t\t\t\t 14.

Explanation:

Problem 5: 15 and 36

Step1: Check if hypotenuse is given

We use Pythagorean theorem $a^2 + b^2 = c^2$, where $c$ is hypotenuse. First test if 36 is hypotenuse: $15^2 + b^2 = 36^2$ → $b^2=1296-225=1071$, not a perfect square. So both are legs.

Step2: Calculate hypotenuse

$c=\sqrt{15^2 + 36^2}=\sqrt{225+1296}=\sqrt{1521}=39$
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Problem 6: 70 and 250

Step1: Check if 250 is hypotenuse

$70^2 + b^2=250^2$ → $b^2=62500-4900=57600$

Step2: Calculate missing leg

$b=\sqrt{57600}=240$, this is a leg.
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Problem 7: 45 and 51

Step1: Check if 51 is hypotenuse

$45^2 + b^2=51^2$ → $b^2=2601-2025=576$

Step2: Calculate missing leg

$b=\sqrt{576}=24$, this is a leg.
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Problem 8: 15 and 20

Step1: Check if 20 is hypotenuse

$15^2 + b^2=20^2$ → $b^2=400-225=175$, not perfect square. So both are legs.

Step2: Calculate hypotenuse

$c=\sqrt{15^2+20^2}=\sqrt{225+400}=\sqrt{625}=25$
#

Problem 9: 96 and 100

Step1: Check if 100 is hypotenuse

$96^2 + b^2=100^2$ → $b^2=10000-9216=784$

Step2: Calculate missing leg

$b=\sqrt{784}=28$, this is a leg.
#

Problem 10: 36 and 60

Step1: Check if 60 is hypotenuse

$36^2 + b^2=60^2$ → $b^2=3600-1296=2304$

Step2: Calculate missing leg

$b=\sqrt{2304}=48$, this is a leg.
#

Problem 11

Step1: Use Pythagorean theorem (legs=7)

$x=\sqrt{7^2 + 7^2}$

Step2: Simplify the radical

$x=\sqrt{49+49}=\sqrt{98}=7\sqrt{2}$
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Problem 12

Step1: Use Pythagorean theorem (hypotenuse=9, leg=5)

$x=\sqrt{9^2 - 5^2}$

Step2: Simplify the radical

$x=\sqrt{81-25}=\sqrt{56}=2\sqrt{14}$
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Problem 13

Step1: Use Pythagorean theorem (hypotenuse=12, leg=$6\sqrt{3}$)

$x=\sqrt{12^2 - (6\sqrt{3})^2}$

Step2: Simplify the radical

$x=\sqrt{144-108}=\sqrt{36}=6$
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Problem 14

Step1: Use Pythagorean theorem (legs=$2\sqrt{15}$ and $\sqrt{15}$)

$x=\sqrt{(2\sqrt{15})^2 + (\sqrt{15})^2}$

Step2: Simplify the radical

$x=\sqrt{60+15}=\sqrt{75}=5\sqrt{3}$

Answer:

1. Third side: 39, it is the hypotenuse
2. Third side: 240, it is a leg
3. Third side: 24, it is a leg
4. Third side: 25, it is the hypotenuse
5. Third side: 28, it is a leg
6. Third side: 48, it is a leg
7. $7\sqrt{2}$
8. $2\sqrt{14}$
9. $6$
10. $5\sqrt{3}$