A rectangle 16 cm long and 13 cm wide. Find the angles formed with the sides and a diagonal of the rectangle

Question

A rectangle 16 cm long and 13 cm wide. Find the angles formed with the sides and a diagonal of the rectangle 

Answer 1

Dylan,

 

I am assuming you are looking for a diagonal, and then the angles between this diagonal and each of two adjacent sides, of the rectangle.

First, lets find the length of the diagonal.

 

In any right triangle,  A^2 + B^2 = C^2, where A & B are the two sides of the rectangle. C is the diagonal, or hypotenuse of the triangle.

 

Solving, C = sqrt(13 ^ 2  +  16 ^ 2) => C = sqrt (425) = 20.62

 

Lets call the angle between C and the 16cm side, Theta

 

We know, from trigonometry, that C * cos(theta) = 16cm

Solving, cos(theta) = 16 / 20.62 = 0.776

Then, theta = arc cos (0.776) =  39.1 degrees

 

There are now two ways to compute the angle between C and the 13 cm side. One is to realize that the angle between two sides, of the rectangle, are 90 and subtract 39.1 from this, giving 50.9.

 

The other is to do the same calculation as above, but with the narrower side.

 

If we call the second angle delta, C * cos(delta) = 13.

cos(delta) = 13 / 20.62 = 0.63,   delta = arc cos (0.63) = 50.9

 

Note that we did not use the sin function to compute delta. If we had, we would have been re-calculating theta, and would have gotten the same answer for theta. We would not have gotten delta.

 

Answer 2

If you draw this out, you see that the sides and the diagonal form a right triangle where the legs are 13 and 16.  The legs are the OPPOSITE and ADJACENT sides of the triangle relative to the angles they are looking for.

 

So is x is the angle where the 13 cm side is opposite, tan x = 13 / 16.  Using a calculator you can find that one of the angles is 39.1°.

 

There are two ways to find the other angle.

 

1. Go to the other angle and do the same thing.  tan y = 16 / 13.  y = 50.9°.

 

2. Since the sum of the angles in a triangle is 180°, the sum of the non-right angles is 90.  So the other angle is 90 - 39.1 = 50.9°.