Question Video: Finding the Length of a Side in a Triangle given the Corresponding Side in a Similar Triangle and the Similarity Ratio between Them | Nagwa Question Video: Finding the Length of a Side in a Triangle given the Corresponding Side in a Similar Triangle and the Similarity Ratio between Them | Nagwa

# Question Video: Finding the Length of a Side in a Triangle given the Corresponding Side in a Similar Triangle and the Similarity Ratio between Them Mathematics • First Year of Secondary School

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If π»π· = 17.5 cm, π·π΄ = 11.3 cm, and πΆπ΅ = 70 cm, find the length of π΄πΆ.

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### Video Transcript

If π»π· equals 17.5 centimeters, π·π΄ equals 11.3 centimeters, and πΆπ΅ equals 70 centimeters, find the length of π΄πΆ.

Letβs begin by filling in the given measurements onto the diagram. So we have π»π· is 17.5 centimeters, π·π΄ is 11.3 centimeters, and πΆπ΅ equals 70 centimeters. The length that we want to calculate is π΄πΆ. It might be useful to see if we can establish if the triangle π΄π·π» and π΄πΆπ΅ are similar. Two of the ways that we can show similarity are by using the AA rule or the SSS rule. In the AA rule, we show that there are two pairs of corresponding angles congruent. In the SSS rule, we show that there are three pairs of corresponding sides in proportion.

In this question, we can see that weβre not given enough information about the sides. So letβs see if we can use the AA rule. Weβre not given any angle measurements. But if we look at this angle, π»π΄π·, thereβs a congruent angle to it. And thatβs at the angle πΆπ΄π΅. This is because we have a pair of vertically opposite angles. Looking at the angle π΄π»π· and using the fact that we have a pair of parallel lines and a transversal π»π΅, then the angle π΄π΅πΆ would be congruent to this one. This means that weβve found two pairs of corresponding angles congruent. And itβs sufficient to say that our triangles π΄π»π· and π΄π΅πΆ are similar.

Notice that we couldβve also used the angles π»π·π΄ and π΄πΆπ΅ to show another pair of corresponding congruent angles. In a triangle, knowing that two pairs of corresponding angles are congruent automatically means that the final pair of corresponding angles are also congruent. So now that weβve shown that we have similar triangles, letβs see if we can work out the length of π΄πΆ.

In similar triangles, the sides are in proportion, so letβs see if we can work out this proportion. Weβre given the lengths of πΆπ΅ and π»π·, and these two sides are corresponding. We want to work out the length π΄πΆ, so weβll need to establish which side is corresponding to this one. Well, itβs the length π΄π·. When weβre writing our proportion relationship, we want to make sure that we get our lengths π΄πΆ and π΄π· in the correct place. π΄πΆ is part of the triangle that also includes πΆπ΅. And π΄π· is part of the triangle that includes the length π»π·.

We can now fill in the lengths that we know. πΆπ΅ is 70 centimeters, π»π· is 17.5 centimeters, and π΄π· is 11.3 centimeters. We can take the cross-product to find our missing length for π΄πΆ. This gives us π΄πΆ times 17.5 equals 70 times 11.3. Evaluating the right-hand side gives us 17.5 times π΄πΆ equals 791. Dividing both sides by 17.5 gives us that π΄πΆ equals 45.2. And the units here will be centimeters.

An alternative method of working out could have included that the scale factor from the triangle π΄π·π» to triangle π΄π΅πΆ is four. So we would multiply the lengths by four. Multiplying π΄π·, which is 11.3 centimeters, by four wouldβve given us that π΄πΆ is equal to 45.2 centimeters. Either method would lead us to the answer that π΄πΆ is 45.2 centimeters.

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