Question

find the length of the missing sides of the right triangle in the below figure.
figure of a right triangle with hypotenuse 37, base x, height 3x - 1, and right angle at the intersection of base and height
a. height: 25, base: 12
b. height: 35, base: 12
c. height: 12, base: 35
d. height: 12, base: 25

Explanation:

Step1: Apply Pythagorean theorem

$$x^2 + (3x-1)^2 = 37^2$$

Step2: Expand and simplify equation

$$x^2 + 9x^2 -6x +1 = 1369$$
$$10x^2 -6x -1368 = 0$$
$$5x^2 -3x -684 = 0$$

Step3: Solve quadratic equation

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=5$, $b=-3$, $c=-684$:
$$x=\frac{3\pm\sqrt{9 + 13680}}{10}=\frac{3\pm\sqrt{13689}}{10}=\frac{3\pm117}{10}$$
Take positive root: $x=\frac{3+117}{10}=12$

Step4: Calculate height

$$3x-1=3(12)-1=35$$

Answer:

B. Height: 35, Base: 12