Question

what value of x will make the triangles similar by the sss similarity theorem? x = the first triangle has sides 21, 15, 15. the second triangle has sides x, 20, 20.

Explanation:

Step1: Recall SSS similarity theorem

For two triangles to be similar by SSS (Side - Side - Side) similarity theorem, the ratios of their corresponding sides must be equal. That is, if we have two triangles with side lengths \(a,b,c\) and \(a',b',c'\) respectively, then \(\frac{a}{a'}=\frac{b}{b'}=\frac{c}{c'}\).

In the first triangle, the side lengths are \(15,15,21\). In the second triangle, the side lengths are \(20,20,x\). The sides of length \(15\) in the first triangle correspond to the sides of length \(20\) in the second triangle, and the side of length \(21\) in the first triangle corresponds to the side of length \(x\) in the second triangle.

Step2: Set up the proportion

We set up the proportion using the corresponding sides. The ratio of the equal - length sides is \(\frac{15}{20}\), and the ratio of the remaining sides should be equal to this ratio. So we have the equation \(\frac{21}{x}=\frac{15}{20}\).

Step3: Solve for \(x\)

Cross - multiply the proportion \(\frac{21}{x}=\frac{15}{20}\). Cross - multiplying gives us \(15x = 21\times20\).

First, calculate \(21\times20=420\). So the equation becomes \(15x = 420\).

Then, divide both sides of the equation by \(15\): \(x=\frac{420}{15}\).

Simplify \(\frac{420}{15}\): \(420\div15 = 28\).

Answer:

\(28\)