Question

a large right triangle is going to be a part of a geometric sculpture, as shown below. the hypotenuse will be 26 feet long. the length of one leg of the triangle is 4 feet more than twice the other leg. find the length of each leg, in feet, and separate them with a comma.
(image of a right triangle with legs labeled ( x ) and ( 2x + 4 ), hypotenuse labeled 26)

Explanation:

Step1: Apply Pythagorean theorem

$$x^2 + (2x+4)^2 = 26^2$$

Step2: Expand all terms

$$x^2 + 4x^2 + 16x + 16 = 676$$

Step3: Simplify to quadratic form

$$5x^2 + 16x - 660 = 0$$

Step4: Solve quadratic equation

Use quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, where $a=5$, $b=16$, $c=-660$:
$$x=\frac{-16\pm\sqrt{16^2-4(5)(-660)}}{2(5)}=\frac{-16\pm\sqrt{256+13200}}{10}=\frac{-16\pm\sqrt{13456}}{10}=\frac{-16\pm116}{10}$$

Step5: Select positive solution

Length cannot be negative, so:
$$x=\frac{-16+116}{10}=\frac{100}{10}=10$$

Step6: Calculate second leg

$$2x+4=2(10)+4=24$$

Answer:

10, 24