The triangles 𝐴𝐵𝐶 and 𝐶𝐷𝐸 are similar. Determine which of the following is equivalent to 𝐴𝐵 over 𝐸𝐷. Circle your answer.
If two triangles are similar, then one is just an enlargement of the other. And it means that corresponding pairs of angles in the two triangles — so that’s angles which are in the same position in the two triangles — are equal.
To find the ratio which is equivalent to 𝐴𝐵 over 𝐸𝐷, we need to work out which side on the smaller triangle is corresponding with 𝐴𝐵 and which side on the larger triangle is corresponding with 𝐸𝐷. We can do this by looking at the angles of the two triangles.
First, we note that angle 𝐴𝐶𝐵 is equal to angle 𝐷𝐶𝐸, as they’re vertically opposite angles. These angles formed by the intersection of two straight lines. And we know that vertically opposite angles are equal.
Next, we see that the lines 𝐴𝐵 and 𝐸𝐷 are parallel, which is what’s indicated by these arrows here. This means that the angles 𝐴𝐵𝐶 and 𝐷𝐸𝐶 are alternate angles in parallel lines. They’re enclosed within a Z-shape. Alternate angles are equal, so we have that angle 𝐴𝐵𝐶 equals angle 𝐷𝐸𝐶.
Next, we can deduce that angle 𝐶𝐴𝐵 and angle 𝐶𝐷𝐸 are also equal. And this is because the angle sum in a triangle is always 180 degrees. So if the other two angles between these triangles are equal, then the third angle must also be equal in order to make the sum 180 degrees.
Now that we’ve identified which angles in the two triangles correspond to one another, we can work out which sides correspond to each other by looking at their relative position between the angles. The side 𝐴𝐵 is between the angles marked in pink and orange. So the side that corresponds to this on the other triangle is the side 𝐸𝐷. This means that the ratio we’ve been given in the question of 𝐴𝐵 over 𝐸𝐷 is actually just the ratio of corresponding sides in these triangles, which means we need to look at which of the four options is actually a ratio of corresponding sides.
The side 𝐴𝐶 is between the pink and green angles. So on the other triangle, that corresponds to the side 𝐶𝐷. So the final pair of corresponding sides between the two triangles, which in each case is between the orange and green angles, is 𝐵𝐶 and 𝐶𝐸.
If we look at the four options, we see straight away that 𝐵𝐶 over 𝐶𝐸 uses a pair of corresponding sides. And so this will be equivalent to 𝐴𝐵 over 𝐸𝐷. The other three ratios do not use pairs of corresponding sides. For example, if we look at 𝐵𝐴 over 𝐶𝐷, this uses a pink side on the larger triangle but a green side on the smaller triangle. So none of these three ratios are equivalent to 𝐴𝐵 over 𝐸𝐷. Our answer then is 𝐵𝐶 over 𝐶𝐸.