### Video Transcript

The shape π΄πΆπ· is a triangle. The two marked angles π΄πΈπ΅ and
π΄π·πΆ are equal. π΄π΅ is equal to 6.8
centimeters. π΄πΆ is equal to 8.5
centimeters. π·πΆ is equal to 9.5 centimeters.
a) Find the length of πΈπ΅ 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 canβt just measure
the sides and answer questions that way. Weβre gonna have to use another
method. So the first step Iβm actually
gonna complete is to mark on all the information that we know.

So first of all, what Iβve actually
done is marked on the lengths of the sides we know. So we know that π·πΆ is equal to
9.5, π΄π΅ is equal to 6.8, and π΄πΆ is equal to 8.5 centimeters. What we also know is that we got
angle π·π΄πΆ in both triangles. So therefore, we got angle π·π΄πΆ
in both triangles. Then, we know that our left-hand
angles are the same in both triangles. So therefore, angle π΄π΅πΈ must be
equal to angle π΄πΆπ·. And also, this is because theyβre
corresponding angles because weβve now got a pair of parallel lines.

Okay, great, so weβve now got all
the information on the triangles that we need. So letβs get on and find the length
of πΈπ΅. So the first important thing to
note is that our triangles π΄π΅πΈ and π΄πΆπ· are similar. And theyβre similar because
actually theyβve got three angles that are the same because weβve got the angle
π·π΄πΆ, which is shared. Then, we know that π΄πΈπ΅ is equal
to π΄π·πΆ and we know that π΄π΅πΈ is equal to π΄πΆπ·.

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 weβre 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 weβre actually now at
a position to find πΈπ΅ because πΈπ΅βs corresponding side is the side π·πΆ. So therefore, we can say that the
length of πΈπ΅ is gonna be equal to our scale factor multiplied by π·πΆ, which is
gonna be equal to 0.8 multiplied by 9.5. So therefore, we can say that πΈπ΅
is equal to 7.6 centimeters. Okay, great, we found that letβs
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΄πΈ is equal to
6.2 centimeters. So Iβve now marked this on the
graph. Okay, so letβs use this to actually
find out the length of π΄π·.

Well, the first thing to note is
that π΄π· and π΄πΈ 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΄πΈ. 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΄πΈ 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, weβre done. Weβve actually answered part a and
part b.