# Video: Pack 3 β’ Paper 3 β’ Question 3

Pack 3 β’ Paper 3 β’ Question 3

03:49

### Video Transcript

The shape π΄πΆπ· is a triangle. The two marked angles π΄πΈπ΅ and π΄π·πΆ are equal. π΄π΅ is equal to 6.8 centimeters. π΄πΆ is equal to 8.5 centimeters. π·πΆ is equal to 9.5 centimeters. a) Find the length of πΈπ΅ and b) find the length of π΄π·.

Well, the first thing to know is that the diagram is not drawn to scale. So therefore, we canβt just measure the sides and answer questions that way. Weβre gonna have to use another method. So the first step Iβm actually gonna complete is to mark on all the information that we know.

So first of all, what Iβve actually done is marked on the lengths of the sides we know. So we know that π·πΆ is equal to 9.5, π΄π΅ is equal to 6.8, and π΄πΆ is equal to 8.5 centimeters. What we also know is that we got angle π·π΄πΆ in both triangles. So therefore, we got angle π·π΄πΆ in both triangles. Then, we know that our left-hand angles are the same in both triangles. So therefore, angle π΄π΅πΈ must be equal to angle π΄πΆπ·. And also, this is because theyβre corresponding angles because weβve now got a pair of parallel lines.

Okay, great, so weβve now got all the information on the triangles that we need. So letβs get on and find the length of πΈπ΅. So the first important thing to note is that our triangles π΄π΅πΈ and π΄πΆπ· are similar. And theyβre similar because actually theyβve got three angles that are the same because weβve got the angle π·π΄πΆ, which is shared. Then, we know that π΄πΈπ΅ is equal to π΄π·πΆ and we know that π΄π΅πΈ is equal to π΄πΆπ·.

Okay, so great, we know that our triangles are similar. So therefore, we know that one is going to be an enlargement of the other. So we need to first of all find out the scale factor. Well, we know the scale factor of any enlargement is equal to the new length divided by the original length. Well, the two corresponding lengths we know are 6.8 and 8.5. So therefore, we can say that the new length is gonna be 6.8 because actually weβre gonna go from the bigger triangle to the smaller triangle. And then, that means the original length was 8.5. So therefore, we can say that the scale factor is equal to 6.8 divided by 8.5, which gives us a scale factor of 0.8.

So that means weβre actually now at a position to find πΈπ΅ because πΈπ΅βs corresponding side is the side π·πΆ. So therefore, we can say that the length of πΈπ΅ is gonna be equal to our scale factor multiplied by π·πΆ, which is gonna be equal to 0.8 multiplied by 9.5. So therefore, we can say that πΈπ΅ is equal to 7.6 centimeters. Okay, great, we found that letβs move on to part b.

Now, what we need to do is actually find the length of π΄π·. Well, to actually find this length, the question gives us this extra bit of information. It tells us that π΄πΈ is equal to 6.2 centimeters. So Iβve now marked this on the graph. Okay, so letβs use this to actually find out the length of π΄π·.

Well, the first thing to note is that π΄π· and π΄πΈ are actually corresponding sides on our similar triangles. So therefore, the length of π΄π· multiplied by a scale factor is gonna be equal to the length of π΄πΈ. So therefore, if we actually divide both sides of our equation by the scale factor, we can say that π΄π· is gonna be equal to π΄πΈ divided by our scale factor, which therefore is gonna give us that π΄π· is equal to 6.2 divided by our scale factor, which was 0.8. So therefore, we can say that the length of π΄π· is equal to 7.75 centimeters. And there, weβre done. Weβve actually answered part a and part b.