Question

a certain triangle has a base (x) and a height (y). a new triangle is created by tripling the length of the base and quadrupling the height. how does the area of the second triangle compare to the area of the first?
a. 3 times larger
b. 4 times larger
c. 6 times larger
d. 12 times larger
e. 7 times larger

Explanation:

Step1: Recall area formula for triangle

The area formula of a triangle is $A = \frac{1}{2}bh$, where $b$ is the base and $h$ is the height. Let the base of the first - triangle be $x$ and the height be $y$, so the area of the first triangle $A_1=\frac{1}{2}xy$.

Step2: Find base and height of the second triangle

The base of the second triangle is $3x$ (tripling the base of the first triangle) and the height is $4y$ (quadrupling the height of the first triangle).

Step3: Calculate area of the second triangle

Using the area formula $A=\frac{1}{2}bh$, the area of the second triangle $A_2=\frac{1}{2}(3x)(4y)$. Simplify $A_2$: $A_2=\frac{1}{2}\times3x\times4y = 6xy$.

Step4: Compare the two areas

To find out how many times larger $A_2$ is than $A_1$, divide $A_2$ by $A_1$. $\frac{A_2}{A_1}=\frac{6xy}{\frac{1}{2}xy}$. The $xy$ terms cancel out, and $\frac{6}{\frac{1}{2}}=12$.

Answer:

D. 12 times larger