Question

triangle xyz has vertices at x(3, 2), y(8, 2), and z(3, 6). the triangle will be dilated by a scale factor of 4. what is the difference between the area of triangle xyz and the area of triangle xyz?
a 30 square units
b 40 square units
c 150 square units
d 160 square units

Explanation:

Step1: Calculate base and height of original triangle

The base of $\triangle XYZ$ is the distance between $X(3,2)$ and $Y(8,2)$. Using the distance formula for points with the same $y$-coordinate, base $b = 8 - 3=5$. The height is the distance between $Z(3,6)$ and the line $y = 2$ (where $X$ and $Y$ lie), so height $h=6 - 2 = 4$.

Step2: Calculate area of original triangle

The area formula for a triangle is $A=\frac{1}{2}bh$. For $\triangle XYZ$, $A_{1}=\frac{1}{2}\times5\times4 = 10$ square units.

Step3: Calculate area of dilated triangle

When a figure is dilated by a scale factor $k$, the area of the new figure is $k^{2}$ times the area of the original figure. Here $k = 4$, so $A_{2}=k^{2}\times A_{1}=4^{2}\times10=160$ square units.

Step4: Calculate the difference in areas

The difference $\Delta A=A_{2}-A_{1}=160 - 10=150$ square units.

Answer:

C. 150 square units