Question
if you have two triangles. The first triangle has a base of 4 feet and a a side that is 3 feet. The second triangle is an enlargement of the first triangle. Don’t know the base but the side this is 9 feet. What were the area of the complete completed flag in square feet B
Answers
Answered by
GPT-4o mini
To determine the area of the complete flag made from the two triangles, we first need to calculate the area of the first triangle and then the second triangle.
### Area of the First Triangle
The area \( A \) of a triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, we have the base of the first triangle as 4 feet. We also know one side is 3 feet, but we need to find the height. Since we don't have a right triangle or direct height measurement, let's consider that the side length may be the height if the given dimensions form a right triangle.
Assuming the triangle is a right triangle with a base of 4 feet and a height of 3 feet, the area would be:
\[
A_1 = \frac{1}{2} \times 4 \times 3 = \frac{1}{2} \times 12 = 6 \text{ square feet}
\]
### Area of the Second Triangle
The second triangle is an enlargement of the first triangle. To find the scale factor:
\[
\text{Scale factor} = \frac{\text{side of the second triangle}}{\text{side of the first triangle}} = \frac{9}{3} = 3
\]
The lengths of the sides of the second triangle are scaled by a factor of 3. Consequently, the area scales by the square of the scale factor:
\[
\text{Area scale factor} = \text{Scale factor}^2 = 3^2 = 9
\]
Now we can find the area of the second triangle:
\[
A_2 = A_1 \times \text{Area scale factor} = 6 \times 9 = 54 \text{ square feet}
\]
### Total Area of the Flag
To find the total area of the flag, we sum the areas of both triangles:
\[
\text{Total Area} = A_1 + A_2 = 6 + 54 = 60 \text{ square feet}
\]
Therefore, the total area of the completed flag is **60 square feet (B)**.
### Area of the First Triangle
The area \( A \) of a triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, we have the base of the first triangle as 4 feet. We also know one side is 3 feet, but we need to find the height. Since we don't have a right triangle or direct height measurement, let's consider that the side length may be the height if the given dimensions form a right triangle.
Assuming the triangle is a right triangle with a base of 4 feet and a height of 3 feet, the area would be:
\[
A_1 = \frac{1}{2} \times 4 \times 3 = \frac{1}{2} \times 12 = 6 \text{ square feet}
\]
### Area of the Second Triangle
The second triangle is an enlargement of the first triangle. To find the scale factor:
\[
\text{Scale factor} = \frac{\text{side of the second triangle}}{\text{side of the first triangle}} = \frac{9}{3} = 3
\]
The lengths of the sides of the second triangle are scaled by a factor of 3. Consequently, the area scales by the square of the scale factor:
\[
\text{Area scale factor} = \text{Scale factor}^2 = 3^2 = 9
\]
Now we can find the area of the second triangle:
\[
A_2 = A_1 \times \text{Area scale factor} = 6 \times 9 = 54 \text{ square feet}
\]
### Total Area of the Flag
To find the total area of the flag, we sum the areas of both triangles:
\[
\text{Total Area} = A_1 + A_2 = 6 + 54 = 60 \text{ square feet}
\]
Therefore, the total area of the completed flag is **60 square feet (B)**.