Question
Archer is sewing a triangular flag. He starts with a scale drawing on the left and plans to enlarge the flag to match the picture on the right what will be the area of the completed flag in square feet the triangle on the left has a base of 3 feet And a side of 4 feet. The area on the right has a base of 9 feet in question on the side length.
Answers
GPT-4o mini
To find the area of the completed flag, we can use the relationship between the scale of the triangles. The original triangle has a base of 3 feet, while the enlarged triangle has a base of 9 feet.
First, we need to determine the scale factor between the two triangles. The scale factor (k) can be calculated as follows:
\[
k = \frac{\text{Base of enlarged triangle}}{\text{Base of original triangle}} = \frac{9 \text{ feet}}{3 \text{ feet}} = 3
\]
This means the dimensions of the enlarged triangle are 3 times larger than the original triangle.
Next, the area of a triangle is given by the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
To find the area of the original triangle, we first need to find its height. We know one side of the triangle is 4 feet, but without the height, we cannot directly calculate the area. However, since we are looking for a scale relationship, we just need to recognize that the area will change based on the square of the scaling factor.
The area of the original triangle (not calculated with height) will be denoted as \( A_1 \). The area of the enlarged triangle \( A_2 \) can be found using the scale factor:
\[
A_2 = k^2 \times A_1
\]
Since the scale factor \( k \) is 3, we have:
\[
A_2 = 3^2 \times A_1 = 9 \times A_1
\]
Now, we need to find \( A_1 \). Since we assume that we need to find the height or some other dimension to calculate \( A_1 \), let’s assume an example value for height based on the dimensions provided.
Assuming the original triangle's height can be derived using the Pythagorean theorem (as the side lengths suggest a right triangle). Considering 3 feet as the base and 4 feet as the hypotenuse, we find the height \( h \).
For a right triangle:
- You can use \( h = \sqrt{4^2 - 3^2} = \sqrt{16 - 9} = \sqrt{7} \approx 2.65 \text{ feet} \).
Thus, \( A_1 \approx \frac{1}{2} \times 3 \times 2.65 \approx 3.975 \text{ square feet} \).
Calculating \( A_2 \):
\[
A_2 = 9 \times A_1 \approx 9 \times 3.975 \approx 35.775 \text{ square feet}.
\]
So the area of the completed flag will be approximately **35.78 square feet**.
The exact area may vary based on the original triangle's actual height, but the principle of how scaling affects area will hold: it multiplies by the square of the scale factor.
First, we need to determine the scale factor between the two triangles. The scale factor (k) can be calculated as follows:
\[
k = \frac{\text{Base of enlarged triangle}}{\text{Base of original triangle}} = \frac{9 \text{ feet}}{3 \text{ feet}} = 3
\]
This means the dimensions of the enlarged triangle are 3 times larger than the original triangle.
Next, the area of a triangle is given by the formula:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
To find the area of the original triangle, we first need to find its height. We know one side of the triangle is 4 feet, but without the height, we cannot directly calculate the area. However, since we are looking for a scale relationship, we just need to recognize that the area will change based on the square of the scaling factor.
The area of the original triangle (not calculated with height) will be denoted as \( A_1 \). The area of the enlarged triangle \( A_2 \) can be found using the scale factor:
\[
A_2 = k^2 \times A_1
\]
Since the scale factor \( k \) is 3, we have:
\[
A_2 = 3^2 \times A_1 = 9 \times A_1
\]
Now, we need to find \( A_1 \). Since we assume that we need to find the height or some other dimension to calculate \( A_1 \), let’s assume an example value for height based on the dimensions provided.
Assuming the original triangle's height can be derived using the Pythagorean theorem (as the side lengths suggest a right triangle). Considering 3 feet as the base and 4 feet as the hypotenuse, we find the height \( h \).
For a right triangle:
- You can use \( h = \sqrt{4^2 - 3^2} = \sqrt{16 - 9} = \sqrt{7} \approx 2.65 \text{ feet} \).
Thus, \( A_1 \approx \frac{1}{2} \times 3 \times 2.65 \approx 3.975 \text{ square feet} \).
Calculating \( A_2 \):
\[
A_2 = 9 \times A_1 \approx 9 \times 3.975 \approx 35.775 \text{ square feet}.
\]
So the area of the completed flag will be approximately **35.78 square feet**.
The exact area may vary based on the original triangle's actual height, but the principle of how scaling affects area will hold: it multiplies by the square of the scale factor.