Question

1. right triangle with one leg (sqrt{26}) m, another leg 5 m, hypotenuse (x)
2. right triangle with one leg (x), another leg 8 in, hypotenuse (sqrt{226}) in
3. right triangle with one leg 3.2, another leg 6.5, hypotenuse (x)
4. right triangle with one leg 5.4, another leg 8.8, hypotenuse (x)
5. right triangle with hypotenuse 11, one leg 5.94, another leg (x)
6. right triangle with one leg 8.67, another leg 5.76, hypotenuse (x)

Explanation:

All problems use the Pythagorean theorem: for a right triangle with legs $a, b$ and hypotenuse $c$, $a^2 + b^2 = c^2$.

Problem 7:

Step1: Identify sides (x=hypotenuse)

$a=\sqrt{26}$, $b=5$, $c=x$

Step2: Apply Pythagorean theorem

$$x^2 = (\sqrt{26})^2 + 5^2 = 26 + 25 = 51$$
$$x = \sqrt{51}$$

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Problem 8:

Step1: Identify sides (x=leg)

$a=x$, $b=8$, $c=\sqrt{226}$

Step2: Isolate $x^2$

$$x^2 = (\sqrt{226})^2 - 8^2 = 226 - 64 = 162$$
$$x = \sqrt{162} = 9\sqrt{2}$$

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Problem 9:

Step1: Identify sides (x=hypotenuse)

$a=3.2$, $b=6.5$, $c=x$

Step2: Calculate $x^2$

$$x^2 = 3.2^2 + 6.5^2 = 10.24 + 42.25 = 52.49$$
$$x = \sqrt{52.49} = 7.245 \approx 7.25$$

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Problem 10:

Step1: Identify sides (x=hypotenuse)

$a=5.4$, $b=8.8$, $c=x$

Step2: Calculate $x^2$

$$x^2 = 5.4^2 + 8.8^2 = 29.16 + 77.44 = 106.6$$
$$x = \sqrt{106.6} \approx 10.32$$

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Problem 11:

Step1: Identify sides (x=leg)

$a=x$, $b=5.94$, $c=11$

Step2: Isolate $x^2$

$$x^2 = 11^2 - 5.94^2 = 121 - 35.2836 = 85.7164$$
$$x = \sqrt{85.7164} \approx 9.26$$

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Problem 12:

Step1: Identify sides (x=leg)

$a=x$, $b=5.76$, $c=8.67$

Step2: Isolate $x^2$

$$x^2 = 8.67^2 - 5.76^2 = 75.1689 - 33.1776 = 41.9913$$
$$x = \sqrt{41.9913} \approx 6.48$$

Answer:

1. $\boldsymbol{\sqrt{51}\ \text{m}}$
2. $\boldsymbol{9\sqrt{2}\ \text{in}}$
3. $\boldsymbol{\approx 7.25}$
4. $\boldsymbol{\approx 10.32}$
5. $\boldsymbol{\approx 9.26}$
6. $\boldsymbol{\approx 6.48}$