Question

1. the hypotenuse of a right triangle is 25 meters in length. one leg is 17 meters longer than the other. find the length of each leg.

Explanation:

Step1: Define variables for legs

Let the shorter leg be $x$ meters. Then the longer leg is $x+17$ meters.

Step2: Apply Pythagorean theorem

$$x^2 + (x+17)^2 = 25^2$$

Step3: Expand and simplify equation

$$x^2 + x^2 + 34x + 289 = 625$$
$$2x^2 + 34x - 336 = 0$$
Divide by 2: $x^2 + 17x - 168 = 0$

Step4: Solve quadratic equation

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=1$, $b=17$, $c=-168$:
$$x=\frac{-17\pm\sqrt{17^2-4(1)(-168)}}{2(1)}$$
$$x=\frac{-17\pm\sqrt{289+672}}{2}$$
$$x=\frac{-17\pm\sqrt{961}}{2}$$
$$x=\frac{-17\pm31}{2}$$
Take positive root: $x=\frac{-17+31}{2}=7$

Step5: Find longer leg

Longer leg: $x+17=7+17=24$

Answer:

The lengths of the legs are 7 meters and 24 meters.