WEBVTT
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Given that triangles π΄π΅πΆ and π΄ prime π΅ prime πΆ prime are similar, work out the value of π₯.
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So the key factor here is that these two triangles are similar, which means that they have proportional side lengths.
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The ratio between corresponding pairs of sides is the same.
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So the ratio between π΅πΆ and π΅ prime πΆ prime is the same as the ratio between π΄πΆ and π΄ prime πΆ prime.
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So we have that π΅πΆ over π΅ prime πΆ prime is equal to π΄πΆ over π΄ prime πΆ prime.
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Weβve been given the lengths of all of these sides exactly for the smaller triangle and in terms of the variable π₯ for the larger triangle.
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Letβs substitute in the expressions or the values for each of the sides.
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For the green sides, the hypotenuse of the triangles, we have two π₯ plus one over six.
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And for the pink side, we have π₯ plus three over four.
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Now what weβve done is form an equation which we now want to solve in order to work out the value of π₯.
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From this stage here, the question is purely an algebraic one.
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Now we have a four and a six in the denominator of these fractions.
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So weβll cross multiply.
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This gives four lots of two π₯ plus one is equal to six lots of π₯ plus three.
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Next, we need to expand the brackets on each side of the equation.
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So we have eight π₯ plus four is equal to six π₯ plus 18.
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Now there are π₯s on both sides of this equation.
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So in order to have π₯s only on one side, in this case the left, we need to subtract six π₯ from both sides.
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This leads us to two π₯ plus four is equal to 18.
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Next, we need to subtract four from both sides of the equation.
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And we now have that two π₯ is equal to 14.
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The final step is to divide both sides of the equation by two.
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So this gives us the solution to the equation and our answer to the problem: π₯ is equal to seven.