Chapter 11.7, Problem 11WE

To determine

To Find: The ratio of perimeters and the ratio of area of $\Delta LMNand\Delta ABC$ .

Answer to Problem 11WE

Ratio of perimeter: $1:2$

Ratio of area: $1:4$

Explanation of Solution

Given: $L,M,N$ are the mid points of the $\Delta ABC$ .

  

In triangle $\Delta LMNand\Delta ABC$

We know that the segment that joins the midpoints of two sides of a triangle is half as long as the third side.

Thus

  $\begin{array}{l}LM=\frac{AC}{2}\\ MN=\frac{AB}{2}\\ LN=\frac{BC}{2}\end{array}$

Therefore

The scalar factor is $1:2$ .

Now

Ratio of perimeter is equal to the scalar factor $1:2$ .

Ratio of area

  $a^2:b^2=1^2:2^2=1:4$

Hence ratio of area is $1:4$ .