Question

the diagram shows the lengths of corresponding sides of similar triangles abc and abc. which expression gives the perimeter of abc? (image shows two right triangles, abc with leg a=4, and abc with leg 8, both right-angled at c and c respectively.)

Explanation:

Step1: Find the scale factor

Since the triangles are similar, the ratio of corresponding sides is the scale factor. The side \(a = 4\) in \(\triangle ABC\) corresponds to \(8\) in \(\triangle A'B'C'\). The scale factor \(k=\frac{8}{4} = 2\).

Step2: Assume the sides of \(\triangle ABC\)

Let the sides of \(\triangle ABC\) be \(a = 4\), \(b\), \(c\). Its perimeter \(P_{ABC}=a + b + c=4 + b + c\).

Step3: Calculate the perimeter of \(\triangle A'B'C'\)

For similar triangles, if the scale factor is \(k\), the ratio of perimeters is also \(k\). The perimeter of \(\triangle A'B'C'\) is \(P_{A'B'C'}=k\times P_{ABC}\). Substituting \(k = 2\) and \(P_{ABC}=4 + b + c\), we get \(P_{A'B'C'}=2\times(4 + b + c)=8 + 2b+2c\).

Answer:

\(8 + 2b + 2c\)