Question

if a diagonal splits a square into two right triangles, and the diagonal is $6sqrt{2}$ units long, what is the length of each side of the square? a. $3sqrt{3}$ units b. 8 units c. 6 units d. 12 units in a right triangle with a hypotenuse of 13 units and one leg of 12 units, what is the length of the other leg? a. 3 units b. 6 units c. 10 units d. 5 units

Explanation:

Step1: Apply Pythagorean theorem for square

Let the side - length of the square be $a$. In a square, if the diagonal $d$ is given, by the Pythagorean theorem $d^{2}=a^{2}+a^{2}=2a^{2}$. Given $d = 6\sqrt{2}$, then $(6\sqrt{2})^{2}=2a^{2}$.

Step2: Solve for $a$

$72 = 2a^{2}$, so $a^{2}=36$, and $a = 6$.

Step3: Apply Pythagorean theorem for right - triangle

Let the hypotenuse be $c$, one leg be $a$ and the other leg be $b$. By the Pythagorean theorem $c^{2}=a^{2}+b^{2}$. Given $c = 13$ and $a = 12$, then $b^{2}=c^{2}-a^{2}$.

Step4: Solve for $b$

$b^{2}=13^{2}-12^{2}=169 - 144=25$, so $b = 5$.

Answer:

1. C. 6 units
2. D. 5 units