Question

determine whether the triangle with the given side lengths is a right triangle.

1. 8, 15, 17
2. 20, 21, 28
3. 9, 12, 15
4. 11, 13, 17
5. 5, 64, 65
6. 12, 25, 27

the lengths of two sides of a right triangle are given. find the length of the third side.

1. ( a = 9 ), ( c = 41 )
2. ( a = 40 ), ( c = 58 )
3. ( b = 56 ), ( c = 65 )
4. ( a = 13 ), ( b = 84 )
5. ( a = 16 ), ( b = 63 )

Explanation:

For each set of side lengths, we use the Pythagorean theorem: for a right triangle, $a^2 + b^2 = c^2$, where $c$ is the longest side.
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7. Test 8,15,17

Step1: Square all sides
$8^2=64$, $15^2=225$, $17^2=289$
Step2: Sum smaller squares
$64+225=289$
Step3: Compare to largest square
$289=289$
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8. Test 20,21,28

Step1: Square all sides
$20^2=400$, $21^2=441$, $28^2=784$
Step2: Sum smaller squares
$400+441=841$
Step3: Compare to largest square
$841
eq784$
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9. Test 9,12,15

Step1: Square all sides
$9^2=81$, $12^2=144$, $15^2=225$
Step2: Sum smaller squares
$81+144=225$
Step3: Compare to largest square
$225=225$
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10. Test 11,13,17

Step1: Square all sides
$11^2=121$, $13^2=169$, $17^2=289$
Step2: Sum smaller squares
$121+169=290$
Step3: Compare to largest square
$290
eq289$
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11. Test 5,64,65

Step1: Square all sides
$5^2=25$, $64^2=4096$, $65^2=4225$
Step2: Sum smaller squares
$25+4096=4221$
Step3: Compare to largest square
$4221
eq4225$
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12. Test 12,25,27

Step1: Square all sides
$12^2=144$, $25^2=625$, $27^2=729$
Step2: Sum smaller squares
$144+625=769$
Step3: Compare to largest square
$769
eq729$
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13. Find side $b$, $a=9,c=41$

Step1: Use Pythagorean theorem
$b^2=c^2-a^2$
Step2: Substitute values
$b^2=41^2-9^2=1681-81=1600$
Step3: Solve for $b$
$b=\sqrt{1600}=40$
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14. Find side $b$, $a=40,c=58$

Step1: Use Pythagorean theorem
$b^2=c^2-a^2$
Step2: Substitute values
$b^2=58^2-40^2=3364-1600=1764$
Step3: Solve for $b$
$b=\sqrt{1764}=42$
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15. Find side $a$, $b=56,c=65$

Step1: Use Pythagorean theorem
$a^2=c^2-b^2$
Step2: Substitute values
$a^2=65^2-56^2=4225-3136=1089$
Step3: Solve for $a$
$a=\sqrt{1089}=33$
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17. Find side $c$, $a=13,b=84$

Step1: Use Pythagorean theorem
$c^2=a^2+b^2$
Step2: Substitute values
$c^2=13^2+84^2=169+7056=7225$
Step3: Solve for $c$
$c=\sqrt{7225}=85$
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18. Find side $c$, $a=16,b=63$

Step1: Use Pythagorean theorem
$c^2=a^2+b^2$
Step2: Substitute values
$c^2=16^2+63^2=256+3969=4225$
Step3: Solve for $c$
$c=\sqrt{4225}=65$

Answer:

1. Yes, it is a right triangle.
2. No, it is not a right triangle.
3. Yes, it is a right triangle.
4. No, it is not a right triangle.
5. No, it is not a right triangle.
6. No, it is not a right triangle.
7. $40$
8. $42$
9. $33$
10. $85$
11. $65$