Question

describe how the given changes to the dimensions of the figure will affect its perimeter or area. what happens to the perimeter of a triangle with side lengths of 3, 5, and 6 units when each side length is decreased by a factor of $\frac{1}{4}$? > move an answer into each space provided. the perimeter of the resulting triangle will be decreased by $\frac{1}{4}$ by 4 by a factor of $\frac{1}{4}$ by a factor of 4

Explanation:

Step1: Calculate original perimeter

The original triangle has side - lengths \(a = 3\), \(b = 5\), and \(c = 6\). The perimeter \(P_{1}\) of a triangle is \(P_{1}=a + b + c\), so \(P_{1}=3 + 5+6=14\) units.

Step2: Calculate new side - lengths

Each side - length is decreased by a factor of \(\frac{1}{4}\). The new side - lengths are \(a_{2}=\frac{3}{4}\), \(b_{2}=\frac{5}{4}\), and \(c_{2}=\frac{6}{4}\).

Step3: Calculate new perimeter

The new perimeter \(P_{2}=a_{2}+b_{2}+c_{2}=\frac{3}{4}+\frac{5}{4}+\frac{6}{4}=\frac{3 + 5+6}{4}=\frac{14}{4}\).

Step4: Analyze the change in perimeter

We can see that \(P_{2}=\frac{1}{4}P_{1}\). So the perimeter is decreased by a factor of \(\frac{1}{4}\).

Answer:

by a factor of \(\frac{1}{4}\)