Video: Using Properties of Similar Triangles to Calculate Lengths and Measures of Corresponding Sides and Angles

Triangles ๐ด๐ต๐ถ and ๐ดโ€ฒ๐ตโ€ฒ๐ถโ€ฒ are similar. Work out the measure of angle ๐‘ฅ. Work out the value of ๐‘ฆ. Work out the value of ๐‘ง.

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Video Transcript

Triangles ๐ด๐ต๐ถ and ๐ด prime ๐ต prime ๐ถ prime are similar. Work out the measure of angle ๐‘ฅ. Work out the value of ๐‘ฆ. Work out the value of ๐‘ง.

In this question, weโ€™re told that our two triangles are similar, which means that corresponding angles are congruent and corresponding sides are in proportion. If we look at this angle at ๐ด prime denoted by the ๐‘ฅ, then we need to work out which angle in triangle ๐ด๐ต๐ถ is corresponding to this one. Sometimes in diagrams, this isnโ€™t always clear, but we can use the order of the letters to help us.

The angle at ๐ด prime will correspond with the angle at ๐ด. We can write this more formally as the angle ๐ถ prime ๐ด prime ๐ต prime is corresponding to the angle at ๐ถ๐ด๐ต, this one in pink. Both of these angles are equal, and theyโ€™re 74.5 degrees. So our answer for the first part of this question is that angle ๐‘ฅ is 74.5 degrees. It can be tempting to think that because the triangle is larger, that the angle must also be larger. But remember that the sum of the angles in a triangle is always 180 degrees.

In the second part of this question, weโ€™re asked to find the length of ๐‘ฆ. Weโ€™ll need to work out the proportion of the lengths or the scale factor that takes us from the smaller triangle to the larger triangle. We can use a given pair of corresponding sides. Here, we have the length ๐ด prime ๐ต prime is five and the length ๐ด๐ต is two. To work out the scale factor from the smaller triangle to the larger triangle, weโ€™ll take our new length of five and divide it by the original length of two. Therefore, if we want to find a length on the longer triangle, we take the corresponding length on the smaller triangle and multiply it by five over two.

So, the length ๐‘ฆ, which we want to calculate on the larger triangle on the line ๐ต prime ๐ถ prime, corresponds with the line ๐ต๐ถ of length three on the smaller triangle. So, we can calculate ๐‘ฆ by multiplying the length three by the scale factor of five over two. Three times five is 15, and 15 over two simplifies to 7.5. And so, our answer to the second part of this question is ๐‘ฆ equals 7.5.

There is an alternative method we couldโ€™ve used to work out the value of ๐‘ฆ. As we know that our triangles are similar, our lengths will all be in the same proportion. Looking at the lengths ๐ด prime ๐ต prime, which is five, and ๐ด๐ต, which is two, we can say that five over two is equal to ๐‘ฆ over three. As these triangles are similar, we know that itโ€™s the same proportion between the lengths five and two as it would be between ๐‘ฆ and three. We can then take the cross product, and so two times ๐‘ฆ is two ๐‘ฆ equals five times three, which is 15. And if two ๐‘ฆ is 15, then ๐‘ฆ is half of that. So, ๐‘ฆ is 7.5, confirming the answer that we found by working out the scale factor.

Letโ€™s take a look at the final part of this question to find the value of ๐‘ง. We know that we go from the smaller triangle to the larger triangle by multiplying the lengths by five over two. But what happens in the reverse direction? In this case, weโ€™d have to perform the inverse operation. We could informally say that we need to divide the lengths by five over two, but scale factors should be given as a multiplier. We can recall that when weโ€™re dividing by a fraction, this is equivalent to multiplying by the fraction flipped. So, our scale factor from the large triangle to the small triangle would be two-fifths.

In order to work out the length of ๐‘ง on the line ๐ด๐ถ, we take the corresponding length ๐ด prime ๐ถ prime, which is four, and multiply it by the scale factor of two-fifths. Four times two is eight, and eight-fifths is equivalent to 1.6. So, our answer for the final part of this question is that ๐‘ง is 1.6.

We couldโ€™ve worked out this final question by using the original scale factor. We wouldโ€™ve set up an equation that said ๐‘ง times five over two equals four. We wouldโ€™ve then rearranged this to find the value of ๐‘ง. This method does track very closely to finding the reverse scale factor however. Both methods would confirm that ๐‘ง is 1.6.

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