Question

if a right triangle has a hypotenuse of 13 units and one leg of 5 units, what is the length of the other leg?
a. 12 units
b. 11 units
c. 10 units
d. 15 units
if a ladder reaches 12 feet up a wall and the base is 5 feet from the wall, how long is the ladder?
a. 10 feet
b. 12.5 feet
c. 14 feet
d. 13 feet

Explanation:

Step1: Apply Pythagorean theorem for first question

Let the hypotenuse be $c = 13$, one leg be $a = 5$, and the other leg be $b$. The Pythagorean theorem is $a^{2}+b^{2}=c^{2}$, so $b=\sqrt{c^{2}-a^{2}}$. Substituting values, we get $b=\sqrt{13^{2}-5^{2}}=\sqrt{169 - 25}=\sqrt{144}=12$.

Step2: Apply Pythagorean theorem for second question

Let the height on the wall be $a = 12$, the base - distance from the wall be $b = 5$, and the length of the ladder be $c$. By the Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}$. Substituting values, $c=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13$.

Answer:

a. 12 units
d. 13 feet