Find the length of line segment 𝐸𝐶.
Before we start looking at the length of 𝐸𝐶, let’s have a look at our diagram and see if we can work out any of the shapes in it. If we look at our shape 𝐷𝐴𝐵𝐶, we can see that there are two parallel sides and two 90-degree angles. So the angle 𝐷𝐴𝐵 and the angle 𝐷𝐶𝐵 must also be 90 degrees, as we would have pairs of supplementary angles. We can conclude, therefore, that our shape 𝐷𝐴𝐶𝐵 is a rectangle. We have been asked to find the length of line segment 𝐸𝐶. But so far, using the fact that 𝐷𝐴𝐶𝐵 is a rectangle isn’t enough information to help us with this. Let’s then have a look at the other shapes in this diagram.
We have a triangle 𝐹𝐷𝐴 and a triangle 𝐹𝐶𝐸. Let’s start by looking at our side length 𝐷𝐹 in our purple Triangle and our side length 𝐹𝐶 in our orange triangle. We can see from our diagram that these two lengths are congruent. That is, line segment 𝐷𝐹 is equal to the line segment 𝐹𝐶. We can see that the measure of angle 𝐹𝐷𝐴 is 90 degrees. And if we look at the angle 𝐹𝐶𝐸 in triangle 𝐹𝐶𝐸, then we can note that this must be 90 degrees since we have a straight line with one 90-degree angle on it. So we can say that the measure of angle 𝐹𝐶𝐸 is 90 degrees and giving us that we have two equivalent angles in our triangles.
Next, if we have a look at the angle 𝐷𝐹𝐴 in triangle 𝐹𝐷𝐴, then we can say that it must be equal to the measure of angle 𝐸𝐹𝐶 in triangle 𝐹𝐶𝐸. Since these are opposite angles. So if we look at the information we’ve just recorded, we can see that we have an equivalent angle and equivalent side and another equivalent angle. This means that we can say that our triangles 𝐹𝐷𝐴 and 𝐹𝐶𝐸 are congruent using the angle side angle congruency criterion.
Let’s see if this helps us workout the length of line segment 𝐸𝐶. We can establish since our triangles are congruent, then our length 𝐸𝐶 in triangle 𝐹𝐶𝐸 is congruent to our length 𝐷𝐴 in triangle 𝐹𝐷𝐴. But what is the length of 𝐷𝐴? Well, if we return to the fact that our shape 𝐷𝐴𝐶𝐵 is a rectangle, we know that opposite sides are equal in length. So the length of 𝐷𝐴 must be equal to the length of 𝐶𝐵, which is 7.5 centimetres. And so we conclude that because 𝐷𝐴 is equal to 7.5 centimetres, then the length of our line segment 𝐸𝐶 is also 7.5 centimetres.