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 choose to cut the paper diagonally. Each piece of cardstock paper measures five 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?(1 point)

Responses

55°
55°

35°
35°

65°
65°

45°
45°

1 answer

To find the angle formed at the base of the triangle after cutting the cardstock paper diagonally, we can use trigonometry.

The triangle has a height \( h = 5 \) inches and a base \( b = 3.5 \) inches. Since we are cutting the triangle diagonally to form two right triangles, we can use the tangent function to find the angle \( \theta \):

\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{\frac{b}{2}} = \frac{5}{\frac{3.5}{2}} = \frac{5}{1.75} \]

Calculating this gives:

\[ \tan(\theta) = \frac{5}{1.75} \approx 2.857 \]

Now, to find the angle \( \theta \), we can use the arctan (inverse tangent) function:

\[ \theta = \tan^{-1}(2.857) \]

Using a calculator or trigonometric tables to find the angle:

\[ \theta \approx 70° \]

However, the angle you are looking for is at the base of the original triangle, not the angle at the top. Since a triangle's three angles add up to 180°, if the apex angle is approximately 70°, we can find the base angle. The base angles will be equal due to the properties of isosceles triangles (the triangle is symmetrical when cut), so we can calculate:

Let the base angles be \( x \):

\[ x + x + 70° = 180° \ 2x = 110° \ x = 55° \]

Thus, the approximate angle formed at the base of the triangle is 55°.

So the answer is:

55°