Question

find the missing length in each right triangle. then determine the slope of the hypotenuse given that the legs of the triangle are vertical and horizontal.
18.
19.
20.
missing length:
slope of hypotenuse:
missing length:
slope of hypotenuse:
missing length:
slope of hypotenuse:

Explanation:

Problem 18

Step1: Apply Pythagorean theorem

Let missing vertical side = $a$. Hypotenuse $c=17$, horizontal side $b=15$.
$$a^2 + b^2 = c^2$$
$$a = \sqrt{c^2 - b^2} = \sqrt{17^2 - 15^2}$$

Step2: Calculate missing length

$$a = \sqrt{289 - 225} = \sqrt{64} = 8$$

Step3: Calculate slope (rise/run)

Slope = $\frac{\text{vertical change}}{\text{horizontal change}} = \frac{8}{15}$

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Problem 19

Step1: Apply Pythagorean theorem

Let missing vertical side = $a$. Hypotenuse $c=\sqrt{116}$, horizontal side $b=10$.
$$a = \sqrt{c^2 - b^2} = \sqrt{(\sqrt{116})^2 - 10^2}$$

Step2: Calculate missing length

$$a = \sqrt{116 - 100} = \sqrt{16} = 4$$

Step3: Calculate slope (rise/run)

Slope = $\frac{\text{vertical change}}{\text{horizontal change}} = \frac{4}{10} = \frac{2}{5}$

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Problem 20

Step1: Apply Pythagorean theorem

Let missing vertical side = $a$. Hypotenuse $c=\sqrt{58}$, horizontal side $b=3$.
$$a = \sqrt{c^2 - b^2} = \sqrt{(\sqrt{58})^2 - 3^2}$$

Step2: Calculate missing length

$$a = \sqrt{58 - 9} = \sqrt{49} = 7$$

Step3: Calculate slope (rise/run)

Slope = $\frac{\text{vertical change}}{\text{horizontal change}} = \frac{7}{3}$

Answer:

Problem 18

Missing length: $8$
Slope of hypotenuse: $\frac{8}{15}$

Problem 19

Missing length: $4$
Slope of hypotenuse: $\frac{2}{5}$

Problem 20

Missing length: $7$
Slope of hypotenuse: $\frac{7}{3}$