# Video: AQA GCSE Mathematics Higher Tier Pack 1 β’ Paper 1 β’ Question 22

AQA GCSE Mathematics Higher Tier Pack 1 β’ Paper 1 β’ Question 22

03:08

### Video Transcript

The triangles π΄π΅πΆ and πΆπ·πΈ are similar. Determine which of the following is equivalent to π΄π΅ over πΈπ·. 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 β so thatβs angles which are in the same position in the two triangles β are equal.

To find the ratio which is equivalent to π΄π΅ over πΈπ·, we need to work out which side on the smaller triangle is corresponding with π΄π΅ and which side on the larger triangle is corresponding with πΈπ·. We can do this by looking at the angles of the two triangles.

First, we note that angle π΄πΆπ΅ is equal to angle π·πΆπΈ, as theyβ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 π΄π΅ and πΈπ· are parallel, which is whatβs indicated by these arrows here. This means that the angles π΄π΅πΆ and π·πΈπΆ are alternate angles in parallel lines. Theyβre enclosed within a Z-shape. Alternate angles are equal, so we have that angle π΄π΅πΆ equals angle π·πΈπΆ.

Next, we can deduce that angle πΆπ΄π΅ and angle πΆπ·πΈ 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 weβ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 π΄π΅ is between the angles marked in pink and orange. So the side that corresponds to this on the other triangle is the side πΈπ·. This means that the ratio weβve been given in the question of π΄π΅ over πΈπ· 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 π΄πΆ is between the pink and green angles. So on the other triangle, that corresponds to the side πΆπ·. So the final pair of corresponding sides between the two triangles, which in each case is between the orange and green angles, is π΅πΆ and πΆπΈ.

If we look at the four options, we see straight away that π΅πΆ over πΆπΈ uses a pair of corresponding sides. And so this will be equivalent to π΄π΅ over πΈπ·. The other three ratios do not use pairs of corresponding sides. For example, if we look at π΅π΄ over πΆπ·, 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 π΄π΅ over πΈπ·. Our answer then is π΅πΆ over πΆπΈ.