Question

similar figures and area quick check
kierra draws a triangle with a base of 10 cm and a height of 15 cm. she dilates the triangle using a scale factor of \\(\frac{4}{5}\\). then, kierra finds the difference between the two areas by subtracting. how much greater is the area of the original triangle than the area of the dilated triangle?
(1 point)
\\(\bigcirc\\) \\(75\\,\text{cm}^2\\)
\\(\bigcirc\\) \\(48\\,\text{cm}^2\\)
\\(\bigcirc\\) \\(27\\,\text{cm}^2\\)
\\(\bigcirc\\) \\(54\\,\text{cm}^2\\)

Explanation:

Step1: Calculate area of original triangle

The formula for the area of a triangle is $A = \frac{1}{2}bh$, where $b$ is the base and $h$ is the height. For the original triangle, $b = 10$ cm and $h = 15$ cm. So, $A_{original} = \frac{1}{2} \times 10 \times 15$.
$A_{original} = 5 \times 15 = 75$ $cm^2$.

Step2: Find dimensions of dilated triangle

The scale factor is $\frac{4}{5}$. So, the new base $b_{dilated} = 10 \times \frac{4}{5} = 8$ cm and the new height $h_{dilated} = 15 \times \frac{4}{5} = 12$ cm.

Step3: Calculate area of dilated triangle

Using the area formula for a triangle, $A_{dilated} = \frac{1}{2} \times 8 \times 12$.
$A_{dilated} = 4 \times 12 = 48$ $cm^2$.

Step4: Find the difference in areas

Subtract the area of the dilated triangle from the area of the original triangle: $75 - 48 = 27$ $cm^2$.

Answer:

27 cm²