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Triangles π΄π΅πΆ and π΄ prime π΅ prime πΆ prime are similar.
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Work out the value of π₯.
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Work out the value of π¦.
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In this question, we have two triangles which weβre told are similar.
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We should remember that when we have similar triangles, this means that corresponding pairs of angles are congruent or equal and corresponding pairs of sides are in the same proportion.
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We can use this fact to help us work out the values of π₯ and π¦.
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So letβs start with the first question to find the value of π₯.
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When weβre working with similar shapes, weβre usually given or can easily calculate the length of two corresponding sides.
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In this diagram, weβre given the lengths of two corresponding sides π΄π΅ and π΄ prime π΅ prime.
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These lengths are five and three.
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We could write this proportion as π΄π΅ over π΄ prime π΅ prime.
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As we need to find the length of π₯ which is on the line segment π΄ prime πΆ prime, then the corresponding side will be the line segment π΄πΆ.
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So the proportion of π΄π΅ over π΄ prime π΅ prime must be equal to the proportion of π΄πΆ over π΄ prime πΆ prime.
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Notice that we couldβve written this proportionality statement with the numerators and denominators both switched.
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But now all we need to do is fill in the values of the lengths that weβre given.
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So we have five over three is equal to 10 over π₯.
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To solve for π₯, we can begin by taking the cross product.
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And this gives us that five π₯ is equal to 30.
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And then dividing both sides by five, we would get that π₯ must be equal to six.
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And so thatβs the answer for the first part of the question.
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Letβs now look at the second part of this question to find the value of π¦.
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This triangle will still have the ratio of sides, the same as the proportion π΄π΅ over π΄ prime π΅ prime.
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But this time, letβs look at the other two corresponding sides.
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These sides are π΅πΆ and π΅ prime πΆ prime.
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Because we have side π΄π΅ on the numerator, then side π΅πΆ must also be on the numerator, as itβs part of the same triangle.
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So π΄π΅ over π΄ prime π΅ prime must be equal to π΅πΆ over π΅ prime πΆ prime.
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When we fill in the values, weβll have five-thirds on the left-hand side.
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π΅πΆ is equal to seven, and π΅ prime πΆ prime is equal to π¦.
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When we take the cross product at this time, weβll have five times π¦, which is five π¦, is equal to three times seven, which is 21.
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Dividing both sides of this equation by five gives us that π¦ is equal to 21 over five.
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Weβve now fully answered the question.
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π₯ is equal to six, and π¦ is equal to 21 over five.
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Both of these values are lengths, so the units of these would be length units.