Video: Pack 3 β€’ Paper 3 β€’ Question 3

Pack 3 β€’ Paper 3 β€’ Question 3

03:49

Video Transcript

The shape 𝐴𝐢𝐷 is a triangle. The two marked angles 𝐴𝐸𝐡 and 𝐴𝐷𝐢 are equal. 𝐴𝐡 is equal to 6.8 centimeters. 𝐴𝐢 is equal to 8.5 centimeters. 𝐷𝐢 is equal to 9.5 centimeters. a) Find the length of 𝐸𝐡 and b) find the length of 𝐴𝐷.

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

So first of all, what I’ve actually done is marked on the lengths of the sides we know. So we know that 𝐷𝐢 is equal to 9.5, 𝐴𝐡 is equal to 6.8, and 𝐴𝐢 is equal to 8.5 centimeters. What we also know is that we got angle 𝐷𝐴𝐢 in both triangles. So therefore, we got angle 𝐷𝐴𝐢 in both triangles. Then, we know that our left-hand angles are the same in both triangles. So therefore, angle 𝐴𝐡𝐸 must be equal to angle 𝐴𝐢𝐷. And also, this is because they’re corresponding angles because we’ve now got a pair of parallel lines.

Okay, great, so we’ve now got all the information on the triangles that we need. So let’s get on and find the length of 𝐸𝐡. So the first important thing to note is that our triangles 𝐴𝐡𝐸 and 𝐴𝐢𝐷 are similar. And they’re similar because actually they’ve got three angles that are the same because we’ve got the angle 𝐷𝐴𝐢, which is shared. Then, we know that 𝐴𝐸𝐡 is equal to 𝐴𝐷𝐢 and we know that 𝐴𝐡𝐸 is equal to 𝐴𝐢𝐷.

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 we’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 we’re actually now at a position to find 𝐸𝐡 because 𝐸𝐡’s corresponding side is the side 𝐷𝐢. So therefore, we can say that the length of 𝐸𝐡 is gonna be equal to our scale factor multiplied by 𝐷𝐢, which is gonna be equal to 0.8 multiplied by 9.5. So therefore, we can say that 𝐸𝐡 is equal to 7.6 centimeters. Okay, great, we found that let’s move on to part b.

Now, what we need to do is actually find the length of 𝐴𝐷. Well, to actually find this length, the question gives us this extra bit of information. It tells us that 𝐴𝐸 is equal to 6.2 centimeters. So I’ve now marked this on the graph. Okay, so let’s use this to actually find out the length of 𝐴𝐷.

Well, the first thing to note is that 𝐴𝐷 and 𝐴𝐸 are actually corresponding sides on our similar triangles. So therefore, the length of 𝐴𝐷 multiplied by a scale factor is gonna be equal to the length of 𝐴𝐸. So therefore, if we actually divide both sides of our equation by the scale factor, we can say that 𝐴𝐷 is gonna be equal to 𝐴𝐸 divided by our scale factor, which therefore is gonna give us that 𝐴𝐷 is equal to 6.2 divided by our scale factor, which was 0.8. So therefore, we can say that the length of 𝐴𝐷 is equal to 7.75 centimeters. And there, we’re done. We’ve actually answered part a and part b.

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