Question
A rectangle has side of 170 & 130. What is the angle between the diagonals
Answers
GPT-4o mini
To find the angle between the diagonals of a rectangle, we can use the properties of the rectangle and some trigonometry.
1. **Identify the sides**: The rectangle has a width \( a = 130 \) and a height \( b = 170 \).
2. **Calculate the lengths of the diagonals**: The length \( d \) of the diagonals in a rectangle can be calculated using the Pythagorean theorem:
\[
d = \sqrt{a^2 + b^2} = \sqrt{130^2 + 170^2}
\]
Calculating further:
\[
= \sqrt{16900 + 28900} = \sqrt{45800}
\]
\[
= \sqrt{45800} \approx 213.97
\]
3. **Determine the angle between the diagonals**: In a rectangle, the diagonals bisect each other and are of equal length. Therefore, the angle \( \theta \) between the diagonals can be found using the tangent function, knowing that the diagonals split the rectangle into two congruent right triangles.
The diagonals \( d_1 \) and \( d_2 \) form an angle at the center, and we can use the arcsine or arccosine function to find that angle using the triangle formed by half the sides of the rectangle:
\[
\tan\left(\frac{\theta}{2}\right) = \frac{b}{a} = \frac{170}{130}
\]
Now calculating \( \frac{b}{a} \):
\[
\tan\left(\frac{\theta}{2}\right) = \frac{170}{130} = \frac{17}{13} \approx 1.3077
\]
Taking the arctangent:
\[
\frac{\theta}{2} = \tan^{-1}(1.3077) \approx 52.43^\circ
\]
Therefore, to find \( \theta \):
\[
\theta \approx 2 \times 52.43^\circ \approx 104.86^\circ
\]
Thus, the angle between the diagonals of the rectangle is approximately \( 104.86^\circ \).
1. **Identify the sides**: The rectangle has a width \( a = 130 \) and a height \( b = 170 \).
2. **Calculate the lengths of the diagonals**: The length \( d \) of the diagonals in a rectangle can be calculated using the Pythagorean theorem:
\[
d = \sqrt{a^2 + b^2} = \sqrt{130^2 + 170^2}
\]
Calculating further:
\[
= \sqrt{16900 + 28900} = \sqrt{45800}
\]
\[
= \sqrt{45800} \approx 213.97
\]
3. **Determine the angle between the diagonals**: In a rectangle, the diagonals bisect each other and are of equal length. Therefore, the angle \( \theta \) between the diagonals can be found using the tangent function, knowing that the diagonals split the rectangle into two congruent right triangles.
The diagonals \( d_1 \) and \( d_2 \) form an angle at the center, and we can use the arcsine or arccosine function to find that angle using the triangle formed by half the sides of the rectangle:
\[
\tan\left(\frac{\theta}{2}\right) = \frac{b}{a} = \frac{170}{130}
\]
Now calculating \( \frac{b}{a} \):
\[
\tan\left(\frac{\theta}{2}\right) = \frac{170}{130} = \frac{17}{13} \approx 1.3077
\]
Taking the arctangent:
\[
\frac{\theta}{2} = \tan^{-1}(1.3077) \approx 52.43^\circ
\]
Therefore, to find \( \theta \):
\[
\theta \approx 2 \times 52.43^\circ \approx 104.86^\circ
\]
Thus, the angle between the diagonals of the rectangle is approximately \( 104.86^\circ \).