Question
mike draws a triangle and dilates it by a scale factor of 2.5 if the triangles original dimensions are 4 inches high and 6 inches along its base, what is the area in square inches of mikes dilated triangle
Answers
GPT-4o mini
To find the area of Mike's dilated triangle, we first need to calculate the area of the original triangle and then apply the scale factor for the dilation.
1. **Calculate the area of the original triangle**:
The formula for the area of a triangle is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Given that the height is 4 inches and the base is 6 inches, the area of the original triangle is:
\[
\text{Area} = \frac{1}{2} \times 6 \, \text{inches} \times 4 \, \text{inches} = \frac{1}{2} \times 24 \, \text{square inches} = 12 \, \text{square inches}
\]
2. **Determine the scale factor for the area**:
When a shape is dilated by a scale factor \( k \), the area scales by a factor of \( k^2 \). Here, the scale factor is 2.5, so the area will increase by a factor of:
\[
(2.5)^2 = 6.25
\]
3. **Calculate the area of the dilated triangle**:
The area of the dilated triangle is:
\[
\text{Dilated Area} = \text{Original Area} \times (k^2)
\]
Thus,
\[
\text{Dilated Area} = 12 \, \text{square inches} \times 6.25 = 75 \, \text{square inches}
\]
Therefore, the area of Mike's dilated triangle is **75 square inches**.
1. **Calculate the area of the original triangle**:
The formula for the area of a triangle is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Given that the height is 4 inches and the base is 6 inches, the area of the original triangle is:
\[
\text{Area} = \frac{1}{2} \times 6 \, \text{inches} \times 4 \, \text{inches} = \frac{1}{2} \times 24 \, \text{square inches} = 12 \, \text{square inches}
\]
2. **Determine the scale factor for the area**:
When a shape is dilated by a scale factor \( k \), the area scales by a factor of \( k^2 \). Here, the scale factor is 2.5, so the area will increase by a factor of:
\[
(2.5)^2 = 6.25
\]
3. **Calculate the area of the dilated triangle**:
The area of the dilated triangle is:
\[
\text{Dilated Area} = \text{Original Area} \times (k^2)
\]
Thus,
\[
\text{Dilated Area} = 12 \, \text{square inches} \times 6.25 = 75 \, \text{square inches}
\]
Therefore, the area of Mike's dilated triangle is **75 square inches**.