Question Video: Finding the Unknown Side Lengths in a Triangle given the Side Lengths of a Similar One | Nagwa Question Video: Finding the Unknown Side Lengths in a Triangle given the Side Lengths of a Similar One | Nagwa

# Question Video: Finding the Unknown Side Lengths in a Triangle given the Side Lengths of a Similar One Mathematics • First Year of Secondary School

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Given that π΄π΅ = 24 cm, π΄π· = 36 cm, π΄πΆ = 18 cm, and πΈπ = 15 cm, find the length of line segment π΄πΈ and π·π.

02:43

### Video Transcript

Given that π΄π΅ equals 24 centimeters, π΄π· equals 36 centimeters, π΄πΆ equals 18 centimeters, and πΈπ equals 15 centimeters, find the length of line segment π΄πΈ and line segment π·π.

Starting by filling in the lengths on the diagram, we can also see that we have three parallel lines marked on this diagram. And we also have two transversals. A transversal is a line which passes through two lines in the same plane at two distinct points. We can use the fact that if three or more parallel lines are cut by two transversals, then they divide the transversals proportionally.

As weβre given the lengths on the line π·π΅, we can use this to work out the proportional relationship between the segments. We can write that the segment π·π΄ over π΄π΅ is equal to the segment π΄πΈ over π΄πΆ. And we can then substitute the values that weβre given for each segment. On the left-hand side, we have the values of 36 over 24, which is equal to π΄πΈ over 18.

To find the missing value for π΄πΈ, we will take the cross product. Noticing that 12 is a factor of both 36 and 24 means that we can simplify the fraction on the left-hand side as three over two. And thatβs equal to π΄πΈ over 18. Taking the cross product then, we have three times 18 is equal to two times π΄πΈ. So 54 equals two times π΄πΈ. Dividing both sides by two will give us that 27 equals π΄πΈ. And therefore, π΄πΈ is equal to 27 centimeters. And we found our first missing length.

To find the next missing length of π·π, we can notice that, on the other transversal, the length πΈπ will be the corresponding length. Defining the length π·π as π, we can then write that π over 24 is equal to 15 over 18. We can simplify the fraction 15 over 18 to give us π over 24 equals five-sixths. We can then take the cross product. So π times six or six π is equal to 24 times five. And simplifying, we have six π equals 120. So π equals 20. And given that we define the length π·π as π, then we have π·π equals 20 centimeters.

So our final answer is π΄πΈ equals 27 centimeters. π·π equals 20 centimeters.

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