Triangles 𝐴𝐵𝐶 and 𝐴𝐷𝐸 are mathematically similar. 𝐴𝐶 is equal to five, 𝐴𝐵 is equal to 𝑦, 𝐵𝐷 is equal to six, 𝐷𝐸 is equal to 12, 𝐶𝐸 is equal to 𝑥, and 𝐵𝐶 is equal to eight. Work out the values of 𝑥 and 𝑦.
As the two triangles are similar, their corresponding lengths will have the same scale factor. For the purposes of this question, we’ll call the scale factor 𝑘. If we firstly considered the corresponding length 𝐵𝐶 and 𝐷𝐸, then eight multiplied by 𝑘 the scale factor is equal to 12. Dividing both sides of this equation by eight gives us 𝑘 is equal to 12 over eight. Simplifying this fraction by dividing the top and bottom by four gives us 𝑘 is equal to three over two. This means that the scale factor is three over two or 1.5.
In order to calculate the value of 𝑥, we need to consider the corresponding lengths 𝐴𝐶 and 𝐴𝐸. Five multiplied by three over two is equal to 𝑥 plus five. This is because the length of 𝐴𝐶 is five and the length of 𝐴𝐸 is 𝑥 plus five. Multiplying the left-hand side gives us 15 over two. Multiplying both sides of this equation by two gives us 15 is equal to two 𝑥 plus 10. Subtracting 10 from both sides of this equation gives us two 𝑥 is equal to five. Dividing both sides by two gives us a value of 𝑥 of five over two or 2.5.
In order to work out the value of 𝑦, we need to consider the corresponding lengths 𝐴𝐵 and 𝐴𝐷. The length of 𝐴𝐵 is 𝑦 and the length of 𝐴𝐷 is 𝑦 plus six. Therefore, 𝑦 multiplied by our scale factor three over two is equal to 𝑦 plus six. Simplifying the left-hand side gives us three 𝑦 over two. Multiplying both sides of this equation by two gives us three 𝑦 is equal to two 𝑦 plus 12. And finally, subtracting two 𝑦 from both sides of this equation gives us a value for 𝑦 equal to six.
If the two triangles 𝐴𝐵𝐶 and 𝐴𝐷𝐸 are mathematically similar, then 𝑥 equals five over two and 𝑦 equals six.