Question
the base of a rectangle measures 20 ft and the altitude measures 14 feet find the measure of the angle that the diagonal makes with the base
Answers
GPT-4o mini
To find the angle that the diagonal of a rectangle makes with the base, we can use trigonometry.
Let's denote:
- the base \( b = 20 \) ft
- the altitude (height) \( h = 14 \) ft
The diagonal \( d \) of the rectangle can be calculated using the Pythagorean theorem:
\[
d = \sqrt{b^2 + h^2} = \sqrt{(20)^2 + (14)^2} = \sqrt{400 + 196} = \sqrt{596} \approx 24.42 \text{ ft}
\]
Now, to find the angle \( \theta \) that the diagonal makes with the base, we can use the tangent function. The tangent of angle \( \theta \) is the ratio of the opposite side (the height) to the adjacent side (the base):
\[
\tan(\theta) = \frac{h}{b} = \frac{14}{20} = 0.7
\]
Now, we will calculate the angle \( \theta \):
\[
\theta = \tan^{-1}(0.7)
\]
Using a calculator,
\[
\theta \approx 34.99^\circ
\]
Therefore, the measure of the angle that the diagonal makes with the base is approximately \( 35^\circ \).
Let's denote:
- the base \( b = 20 \) ft
- the altitude (height) \( h = 14 \) ft
The diagonal \( d \) of the rectangle can be calculated using the Pythagorean theorem:
\[
d = \sqrt{b^2 + h^2} = \sqrt{(20)^2 + (14)^2} = \sqrt{400 + 196} = \sqrt{596} \approx 24.42 \text{ ft}
\]
Now, to find the angle \( \theta \) that the diagonal makes with the base, we can use the tangent function. The tangent of angle \( \theta \) is the ratio of the opposite side (the height) to the adjacent side (the base):
\[
\tan(\theta) = \frac{h}{b} = \frac{14}{20} = 0.7
\]
Now, we will calculate the angle \( \theta \):
\[
\theta = \tan^{-1}(0.7)
\]
Using a calculator,
\[
\theta \approx 34.99^\circ
\]
Therefore, the measure of the angle that the diagonal makes with the base is approximately \( 35^\circ \).