Using the diagonal of a rectangle to find an angle

Expert intro: This is a rectangle-and-diagonal trigonometry problem. The key is to form a right triangle and compare the two side lengths.

Detailed walkthrough

Form the right triangle

Petra walked diagonally across a rectangular schoolyard that is $45\text{ m}$ by $65\text{ m}$. The diagonal creates a right triangle with legs 45 and 65.

The question asks for the angle with respect to the shorter side, so we measure the angle from the 45 m side.

Use tangent to find the angle

For the angle $\theta$ measured from the shorter side,

$\tan\theta = \frac{65}{45}.$

So

$\theta = \tan^{-1}\left(\frac{65}{45}\right).$

Using a calculator,

$\theta \approx 55.3^\circ.$

Rounded to the nearest degree,

$\boxed{55^\circ}.$

Check the interpretation

Because the diagonal is closer to the longer side, the angle with the shorter side should be greater than $45^\circ$, which matches the result. If you measured the angle from the longer side instead, you would get the complement, about $35^\circ$. The wording of the question matters here, so always identify which side the angle is referenced to before choosing the trigonometric ratio.

💡 Pitfall guide

A very common mistake is to reverse the ratio and compute $\tan^{-1}(45/65)$, which gives the angle with the longer side instead of the shorter side. Another problem is ignoring that the diagonal is the hypotenuse and trying to use cosine without first identifying the reference angle. The safest approach is to draw the right triangle, label the given sides, and state clearly which side the angle is measured from.

🔄 Real-world variant

If the question asked for the angle with respect to the longer side, the answer would be the complement of $55^\circ$, which is $35^\circ$ to the nearest degree. If the yard were $40\text{ m}$ by $60\text{ m}$, the same method would give $\tan^{-1}(60/40)\approx 56^\circ$ measured from the shorter side. The strategy stays the same: identify the reference side and use tangent on the two legs.

🔍 Related terms

diagonal of rectangle, reference angle, inverse tangent