Question Video: Finding the Length of a Projection of a Right-Angled Triangle Side on the Straight Line Carrying Another Side Using Pythagoras’s Theorem

Video Transcript

Given that 𝐴𝐵 equals 29, 𝐶𝐵 equals 20, and 𝐶𝐷 equals 35, calculate the length of the projection of line segment 𝐶𝐷 on line 𝐴𝐷.

Before we can calculate the length of this projection, we’ll need to see which line segment this projection is. Line 𝐴𝐷 is our target line. That’s the line upon which the projection will fall. And 𝐶𝐷 is the line segment that we’ll use to create our projection. But in order to create a projection, we’ll need perpendicular lines to our target line. Because we know that line 𝐵𝐶 and line 𝐴𝐷 are parallel, the angle 𝐵𝐶𝐴 is an alternate interior angle to the angle 𝐶𝐴𝐷, which means the line segment 𝐴𝐶 is perpendicular to the line segment 𝐴𝐷.

Once we imagine the light source as the set of all the perpendicular lines to our target line, we can find the endpoints of our projection. Starting at 𝐶, the projection of point 𝐶 onto 𝐴𝐷 would be the point 𝐴 and the projection of point 𝐷 onto 𝐴𝐷 would be itself. And that means the projection of 𝐶𝐷 is going to be equal to 𝐴𝐷. This is the value we want to calculate the length of. To do that, let’s clear out our light source lines and think about what we know about right triangles.

First of all, we know that 𝐴𝐵 has a measure of 29 and 𝐶𝐵 has a measure of 20 and 𝐶𝐷 has a measure of 35. What we have in our figure is two separate right triangles that make up this quadrilateral. And so we remember that we can find side lengths in right triangles using the Pythagorean theorem, where 𝑎 and 𝑏 represent the two smaller sides and 𝑐 represents the hypotenuse. 𝑎 squared plus 𝑏 squared equals 𝑐 squared. To use the Pythagorean theorem, you need at least two of the lengths. Since we don’t know the length of 𝐴𝐶, we can’t find the length of 𝐴𝐷. However, we can use the information about the smaller triangle 𝐴𝐵𝐶 to solve for the side length 𝐴𝐶 first.

When we plug in what we know, we get 20 squared plus 𝐴𝐶 squared equals 29 squared. 400 plus 𝐴𝐶 squared equals 841. So we’ll subtract 400 from both sides of the equation, and we’ll get 𝐴𝐶 squared equals 441. After that, we’ll take the square root of both sides. We’re only interested in the positive square root since we’re dealing with distance, and so we see that 𝐴𝐶 equals 21. Now that we know that 𝐴𝐶 is 21, we know two distances in our right triangle and we’ll be able to find the side length of our third distance 𝐴𝐷. But we’ll need to set up the Pythagorean theorem for a second time.

This time, we’ll have 21 squared plus 𝐴𝐷 squared equals 35 squared. 441 plus 80 squared equals 1225. And so we subtract 441 from both sides, and we’ll get 𝐴𝐷 squared equals 784. So we take the square root of both sides. Again, we’re only interested in the positive square root of 784, which is 28. The line segment 𝐴𝐷 will be equal to 28. The line segment 𝐴𝐷 is the projection of the line segment 𝐶𝐷 onto line 𝐴𝐷, and it has a measure of 28.