Question

the pythagorean theorem
do the following lengths form a right triangle?
1)
2)
3)
4)

1. $a = 6.4$, $b = 12$, $c = 12.2$
2. $a = 2.1$, $b = 7.2$, $c = 7.5$

find each missing length to the nearest tenth.
7)
8)
9)
10)
11)
12)

Explanation:

For problems 1-6: Check right triangle via Pythagorean theorem ($a^2 + b^2 = c^2$, where $c$ is the longest side)

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1) Step1: Square all side lengths

$6^2=36$, $8^2=64$, $9^2=81$

1) Step2: Sum smaller squares, compare to largest

$36+64=100
eq 81$
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2) Step1: Square all side lengths

$5^2=25$, $12^2=144$, $13^2=169$

2) Step2: Sum smaller squares, compare to largest

$25+144=169 = 169$
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3) Step1: Square all side lengths

$6^2=36$, $8^2=64$, $10^2=100$

3) Step2: Sum smaller squares, compare to largest

$36+64=100 = 100$
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4) Step1: Square all side lengths

$3^2=9$, $4^2=16$, $5^2=25$

4) Step2: Sum smaller squares, compare to largest

$9+16=25 = 25$
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5) Step1: Square all side lengths

$6.4^2=40.96$, $12^2=144$, $12.2^2=148.84$

5) Step2: Sum smaller squares, compare to largest

$40.96+144=184.96
eq 148.84$
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6) Step1: Square all side lengths

$2.1^2=4.41$, $7.2^2=51.84$, $7.5^2=56.25$

6) Step2: Sum smaller squares, compare to largest

$4.41+51.84=56.25 = 56.25$
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For problems 7-12: Find missing side via Pythagorean theorem

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7) Step1: Solve for missing leg $x$

$x=\sqrt{8^2 - 4^2}$

7) Step2: Calculate value to nearest tenth

$x=\sqrt{64-16}=\sqrt{48}\approx6.9$
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8) Step1: Solve for missing hypotenuse $x$

$x=\sqrt{6^2 + 3^2}$

8) Step2: Calculate value to nearest tenth

$x=\sqrt{36+9}=\sqrt{45}\approx6.7$
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9) Step1: Solve for missing leg $x$

$x=\sqrt{10^2 - 7^2}$

9) Step2: Calculate value to nearest tenth

$x=\sqrt{100-49}=\sqrt{51}\approx7.1$
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10) Step1: Solve for missing hypotenuse $x$

$x=\sqrt{3^2 + 7^2}$

10) Step2: Calculate value to nearest tenth

$x=\sqrt{9+49}=\sqrt{58}\approx7.6$
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11) Step1: Solve for missing leg $x$

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

11) Step2: Calculate value to nearest tenth

$x=\sqrt{49-4}=\sqrt{45}\approx6.7$
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12) Step1: Solve for missing leg $x$

$x=\sqrt{6^2 - 2^2}$

12) Step2: Calculate value to nearest tenth

$x=\sqrt{36-4}=\sqrt{32}\approx5.7$

Answer:

1. No, the lengths do not form a right triangle.
2. Yes, the lengths form a right triangle.
3. Yes, the lengths form a right triangle.
4. Yes, the lengths form a right triangle.
5. No, the lengths do not form a right triangle.
6. Yes, the lengths form a right triangle.
7. 6.9
8. 6.7
9. 7.1
10. 7.6
11. 6.7
12. 5.7