Consider the following diagram. Given that 𝐴𝐸 is parallel to 𝐵𝐷, prove that the triangles 𝐴𝐶𝐸 and 𝐵𝐶𝐷 are
similar. You must justify every stage of your working out.
The key word here is “prove.” A proof is a set of statements that show that something is always true. This is a geometric proof, meaning we need to use the geometrical rules we know to
prove the statement that triangles 𝐴𝐶𝐸 and 𝐵𝐶𝐷 are similar. Remember two triangles are similar if that angles are identical. So we need to show that each triangle has the same angles.
First, we can see that the two triangles share an angle. Angle 𝐵𝐶𝐷 is equal to angle 𝐴𝐶𝐸 because it’s a shared angle. We are told that the lines 𝐵𝐷 and 𝐴𝐸 are parallel. We therefore know that angle 𝐵𝐷𝐶 is equal to angle 𝐴𝐸𝐶 because corresponding
angles are equal. Those are the ones that look like the letter F. But remember we do have to use the correct mathematical language and the full
sentence in our exam. It’s not enough just to write corresponding angles. We have to write corresponding angles are equal.
Similarly, angle 𝐶𝐵𝐷 is equal to angle 𝐶𝐴𝐸 because corresponding angles are
equal. We must also remember to include a conclusion after our proof. Here, we say that the angles are equal by writing AAA, standing for angle, angle,
angle. Triangles 𝐴𝐶𝐸 and 𝐵𝐶𝐷 are therefore similar.
Calculate the length of 𝐶𝐷. Once we know that the two triangles are similar, we also know that one must be an
enlargement of the other. This means we need to find the scale factor of the enlargement. To find the scale factor, we divide the enlarged length by its corresponding original
length. Here, the enlarged length is eight and its corresponding old length is two. Eight divided by two gives us a scale factor of four.
Usually, we would now be able to multiply or divide to find the length we’re
interested in, here we don’t know the height of the smaller triangle. So we’re going to need to form an equation. Let’s start by calling the height of the smaller triangle 𝑥. The height of the larger triangle is therefore 𝑥 plus five. We know it is four times larger than the height of the smaller triangle. So 𝑥 plus five is equal to four lots of 𝑥.
To solve, we can start by subtracting 𝑥 from both sides and then dividing by
three. Always ensure to leave your answer as a fraction in its simplest form. The length of 𝐶𝐷 is five out of three centimeters.