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Triangles π΄π΅πΆ and π΄π·πΈ are mathematically similar.
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π΄πΆ is equal to five, π΄π΅ is equal to π¦, π΅π· is equal to six, π·πΈ is equal to 12, πΆπΈ is equal to π₯, and π΅πΆ is equal to eight.
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Work out the values of π₯ and π¦.
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As the two triangles are similar, their corresponding lengths will have the same scale factor.
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For the purposes of this question, weβll call the scale factor π.
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If we firstly considered the corresponding length π΅πΆ and π·πΈ, then eight multiplied by π the scale factor is equal to 12.
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Dividing both sides of this equation by eight gives us π is equal to 12 over eight.
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Simplifying this fraction by dividing the top and bottom by four gives us π is equal to three over two.
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This means that the scale factor is three over two or 1.5.
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In order to calculate the value of π₯, we need to consider the corresponding lengths π΄πΆ and π΄πΈ.
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Five multiplied by three over two is equal to π₯ plus five.
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This is because the length of π΄πΆ is five and the length of π΄πΈ is π₯ plus five.
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Multiplying the left-hand side gives us 15 over two.
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Multiplying both sides of this equation by two gives us 15 is equal to two π₯ plus 10.
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Subtracting 10 from both sides of this equation gives us two π₯ is equal to five.
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Dividing both sides by two gives us a value of π₯ of five over two or 2.5.
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In order to work out the value of π¦, we need to consider the corresponding lengths π΄π΅ and π΄π·.
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The length of π΄π΅ is π¦ and the length of π΄π· is π¦ plus six.
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Therefore, π¦ multiplied by our scale factor three over two is equal to π¦ plus six.
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Simplifying the left-hand side gives us three π¦ over two.
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Multiplying both sides of this equation by two gives us three π¦ is equal to two π¦ plus 12.
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And finally, subtracting two π¦ from both sides of this equation gives us a value for π¦ equal to six.
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If the two triangles π΄π΅πΆ and π΄π·πΈ are mathematically similar, then π₯ equals five over two and π¦ equals six.