Question
A triangular flag has a height of 15 inches and base length of 25 inches. Magnolia makes a scale drawing of the flag in which the base length is 10 inches What is the area of Magnolia's scale drawing? Solve by computing the actual area from the scale drawing. Show your work.
Answers
GPT-4o mini
To find the area of Magnolia's scale drawing of the triangular flag, we first need to calculate the original area of the triangular flag, and then use that to find the area of the scaled-down version.
### Step 1: Calculate the area of the original triangle
The formula for the area \( A \) of a triangle is:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the original flag:
- Base = 25 inches
- Height = 15 inches
Substituting the values into the formula:
\[
A = \frac{1}{2} \times 25 \times 15
\]
Calculating it step-by-step:
\[
A = \frac{1}{2} \times 25 = 12.5
\]
\[
A = 12.5 \times 15 = 187.5 \text{ square inches}
\]
### Step 2: Determine the scale factor from the original to the scale drawing
The original base length is 25 inches and the scale drawing base length is 10 inches. To find the scale factor, we divide the length of the scale drawing by the length of the original flag:
\[
\text{Scale factor} = \frac{10 \text{ inches}}{25 \text{ inches}} = \frac{2}{5}
\]
### Step 3: Calculate the area of the scale drawing
The area of similar shapes scales by the square of the scale factor. Therefore, we need to square the scale factor and then multiply it by the original area:
\[
\text{Scale factor squared} = \left(\frac{2}{5}\right)^2 = \frac{4}{25}
\]
Now, we can find the area of the scale drawing:
\[
\text{Area of scale drawing} = \text{Original area} \times \text{Scale factor squared}
\]
Substituting in the values:
\[
\text{Area of scale drawing} = 187.5 \times \frac{4}{25}
\]
Calculating it:
\[
\text{Area of scale drawing} = \frac{187.5 \times 4}{25} = \frac{750}{25} = 30 \text{ square inches}
\]
### Conclusion
The area of Magnolia's scale drawing of the flag is:
\[
\boxed{30} \text{ square inches}
\]
### Step 1: Calculate the area of the original triangle
The formula for the area \( A \) of a triangle is:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the original flag:
- Base = 25 inches
- Height = 15 inches
Substituting the values into the formula:
\[
A = \frac{1}{2} \times 25 \times 15
\]
Calculating it step-by-step:
\[
A = \frac{1}{2} \times 25 = 12.5
\]
\[
A = 12.5 \times 15 = 187.5 \text{ square inches}
\]
### Step 2: Determine the scale factor from the original to the scale drawing
The original base length is 25 inches and the scale drawing base length is 10 inches. To find the scale factor, we divide the length of the scale drawing by the length of the original flag:
\[
\text{Scale factor} = \frac{10 \text{ inches}}{25 \text{ inches}} = \frac{2}{5}
\]
### Step 3: Calculate the area of the scale drawing
The area of similar shapes scales by the square of the scale factor. Therefore, we need to square the scale factor and then multiply it by the original area:
\[
\text{Scale factor squared} = \left(\frac{2}{5}\right)^2 = \frac{4}{25}
\]
Now, we can find the area of the scale drawing:
\[
\text{Area of scale drawing} = \text{Original area} \times \text{Scale factor squared}
\]
Substituting in the values:
\[
\text{Area of scale drawing} = 187.5 \times \frac{4}{25}
\]
Calculating it:
\[
\text{Area of scale drawing} = \frac{187.5 \times 4}{25} = \frac{750}{25} = 30 \text{ square inches}
\]
### Conclusion
The area of Magnolia's scale drawing of the flag is:
\[
\boxed{30} \text{ square inches}
\]