Chapter 14.3, Problem 20PSC

a.

To determine

To find the ratio of the area of the triangle to the area of the original triangle.

Answer to Problem 20PSC

  $\frac{ar(△DEF)}{ar(△ABC)}=\frac{1}{4}$

Explanation of Solution

Given:

The new triangle is formed by joining the trisection points of the medians.

Calculation:

The sketch will be as follows-

  

  $DECF$ is a parallelogram.

  $\begin{array}{l}ar(△CEF)=ar(△DEF)=ar(△DBE)=ar(△AFD)\\ \Rightarrow 4\times ar(△DEF)=ar(△ABC)\\ \Rightarrow \frac{ar(△DEF)}{ar(△ABC)}=\frac{1}{4}\end{array}$

b.

To determine

To find the relationship of the triangle to the triangle formed by joining the midpoints of the original triangle.

Answer to Problem 20PSC

The area of new triangle will be one fourth of the area of the original triangle.

Explanation of Solution

Given:

The new triangle is formed by joining the trisection points of the medians.

Calculation:

The area of new triangle will be one fourth of the area of the original triangle.

  $ar(△DEF)=\frac{1}{4}ar(△ABC)$