Question
You are making a banner for your cousin's birthday party. You start by cutting out triangle pieces of cardstock paper; to not waste paper, you cut the paper diagonally. Each piece of cardstock measures 5 inches in height with a base of three and a half inches. After cutting, what is the approximate angle formed at the base of the triangle?
Answers
GPT-4o mini
To find the approximate angle formed at the base of the triangle after cutting the cardstock paper diagonally, we can first identify the triangle's dimensions. The triangle has a height of 5 inches and a base of 3.5 inches.
When you cut the triangle diagonally, it creates two right triangles, each with a height of 5 inches and a base which is half of the original base (since cutting diagonally from the apex to the midpoint of the base splits it in half).
Thus, the base of each right triangle becomes:
\[
\text{Base of each right triangle} = \frac{3.5}{2} = 1.75 \text{ inches}
\]
Now we have a right triangle with a height of 5 inches and a base of 1.75 inches. We can use the tangent function to find the angle \( \theta \) formed at the base by using the relationship:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{height}}{\text{base}}
\]
Substituting in the values we have:
\[
\tan(\theta) = \frac{5}{1.75}
\]
Calculating the value of \( \tan(\theta) \):
\[
\tan(\theta) \approx 2.8571
\]
To find the angle \( \theta \), we use the arctangent (inverse tangent):
\[
\theta \approx \tan^{-1}(2.8571)
\]
Using a calculator to find this value:
\[
\theta \approx 71.57^{\circ}
\]
Thus, the approximate angle formed at the base of the triangle is:
\[
\theta \approx 71.57^{\circ}
\]
Therefore, the approximate angle formed at the base of the triangle after cutting is **71.57 degrees**.
When you cut the triangle diagonally, it creates two right triangles, each with a height of 5 inches and a base which is half of the original base (since cutting diagonally from the apex to the midpoint of the base splits it in half).
Thus, the base of each right triangle becomes:
\[
\text{Base of each right triangle} = \frac{3.5}{2} = 1.75 \text{ inches}
\]
Now we have a right triangle with a height of 5 inches and a base of 1.75 inches. We can use the tangent function to find the angle \( \theta \) formed at the base by using the relationship:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{height}}{\text{base}}
\]
Substituting in the values we have:
\[
\tan(\theta) = \frac{5}{1.75}
\]
Calculating the value of \( \tan(\theta) \):
\[
\tan(\theta) \approx 2.8571
\]
To find the angle \( \theta \), we use the arctangent (inverse tangent):
\[
\theta \approx \tan^{-1}(2.8571)
\]
Using a calculator to find this value:
\[
\theta \approx 71.57^{\circ}
\]
Thus, the approximate angle formed at the base of the triangle is:
\[
\theta \approx 71.57^{\circ}
\]
Therefore, the approximate angle formed at the base of the triangle after cutting is **71.57 degrees**.