Question

what is the area, in square centimeters, of the triangle shown? a 45° 24 cm 45° a. 432 b. 216 c. 288 d. 144

Explanation:

Step1: Identify the type of triangle

This is a 45 - 45- 90 right - triangle. In a 45 - 45- 90 triangle, the ratio of the sides is $1:1:\sqrt{2}$. Let the length of each of the legs (the sides adjacent to the 45 - degree angles) be $a$. The hypotenuse $c$ is related to the leg by $c = a\sqrt{2}$. Given $c = 24$ cm, we can find $a$.
Since $c=a\sqrt{2}$ and $c = 24$, then $a=\frac{c}{\sqrt{2}}=\frac{24}{\sqrt{2}}$.

Step2: Rationalize the denominator

Multiply the numerator and denominator of $\frac{24}{\sqrt{2}}$ by $\sqrt{2}$: $a=\frac{24\sqrt{2}}{2}=12\sqrt{2}$ cm.

Step3: Calculate the area of the triangle

The area formula for a triangle is $A=\frac{1}{2}bh$. In a right - triangle, the base $b$ and height $h$ are the two legs. Here, $b = h=a = 12\sqrt{2}$ cm.
So $A=\frac{1}{2}\times(12\sqrt{2})\times(12\sqrt{2})$.
First, $(12\sqrt{2})\times(12\sqrt{2})=12\times12\times\sqrt{2}\times\sqrt{2}=144\times2 = 288$.
Then $A=\frac{1}{2}\times288 = 144$ $cm^{2}$.

Answer:

D. 144