Question

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a triangle has a base of 8 cm and a height of 3 cm. the triangle is dilated by a scale factor of 3. calculate and compare the areas of the dilated triangle and the original triangle.
(2 point)
the original triangle has an area of 24 cm² and the dilated triangle has an area of 36 cm²; therefore, the dilated triangle has an area 1.5 times the size of the original triangle.
the original triangle has an area of 12 cm² and the dilated triangle has an area of 36 cm²; therefore, the dilated triangle has an area 3 times the size of the original triangle.
the original triangle has an area of 12 cm² and the dilated triangle has an area of 108 cm²; therefore, the dilated triangle has an area 9 times the size of the original triangle.
the original triangle has an area of 24 cm² and the dilated triangle has an area of 108 cm²; therefore, the dilated triangle has an area 4.5 times the size of the original triangle.

Explanation:

Step1: Calculate original triangle area

The formula for the area of a triangle is $A = \frac{1}{2} \times base \times height$. Given base = 8 cm and height = 3 cm, so $A_{original} = \frac{1}{2} \times 8 \times 3 = 12 \, cm^2$.

Step2: Find dimensions after dilation

When a figure is dilated by a scale factor $k$, the base and height are multiplied by $k$. Here, $k = 3$, so new base = $8 \times 3 = 24$ cm, new height = $3 \times 3 = 9$ cm.

Step3: Calculate dilated triangle area

Using the area formula again, $A_{dilated} = \frac{1}{2} \times 24 \times 9 = 108 \, cm^2$.

Step4: Compare the areas

To find how many times the dilated area is of the original, divide $A_{dilated}$ by $A_{original}$: $\frac{108}{12} = 9$. So the dilated triangle's area is 9 times the original.

Answer:

The original triangle has an area of $12 \, cm^2$ and the dilated triangle has an area of $108 \, cm^2$; therefore, the dilated triangle has an area 9 times the size of the original triangle.