Question

a and b
are the triangles similar?
scale factor =
geometric shape
original right triangle
a
new right triangle
b
what happens to the perimeter?
what happens to the area?

Explanation:

Step1: Determine if triangles are similar

Check the ratios of corresponding sides. For triangle \(A\) (sides \(3\), \(4\), \(5\)) and triangle \(B\) (sides \(9\), \(12\), \(15\)).
\(\frac{9}{3}=\frac{12}{4}=\frac{15}{5} = 3\). So, the triangles are similar.

Step2: Calculate scale factor

Since \(\frac{\text{side of }B}{\text{side of }A}=3\), the scale factor is \(3\).

Step3: Calculate original triangle \(A\) perimeter

Perimeter of \(A=3 + 4+5=12\).

Step4: Calculate new triangle \(B\) perimeter

Using the scale - factor, perimeter of \(B=12\times3 = 36\).

Step5: Calculate original triangle \(A\) area

Area of \(A=\frac{1}{2}\times3\times4 = 6\).

Step6: Calculate new triangle \(B\) area

Using the scale - factor, area of \(B=\frac{1}{2}\times9\times12=54\). Also, since for similar figures with scale factor \(k\), \(\text{Area ratio}=k^{2}\). Here \(k = 3\), and \(6\times3^{2}=54\).

Answer:

• Are the Triangles Similar? Yes
• Scale Factor: \(3\)
• Original Right Triangle (\(A\)):
• Base: \(3\)
• Height: \(4\)
• Perimeter: \(12\)
• Area: \(6\)
• New Right Triangle (\(B\)):
• Base: \(9\)
• Height: \(12\)
• Perimeter: \(36\)
• Area: \(54\)
• What happens to the Perimeter? The perimeter is multiplied by the scale factor (\(3\))
• What happens to the Area? The area is multiplied by the square of the scale factor (\(3^{2}=9\))