Chapter 7.6, Problem 31WE

To determine

To Find : The value of x for the triangle ABC.

Answer to Problem 31WE

The value of x is 4.

Explanation of Solution

Given information:

  

In $\Delta ABC$ , $AB=8,BC=6,$ and $AC=12$ .

Consider the triangle $\Delta ABC$ .

Here the side length are:

  $AB=8,BC=6,$ and $AC=12$

Consider a point K, three segments are drawn through the point so that length will be x and is parallel to a side of triangle.

Mark the segments as below:

  

Here,

  $\begin{array}{cc}XY& =MN\\ & =QR\\ & =x\end{array}$

Also,

  $\begin{array}{l}XY\parallel AC\\ MN\parallel BC\\ QR\parallel AB\end{array}$

Therefore, the triangle proportionality theorm will be applicable here.

Consider the two triangles $\text{Δ}ABC$ and $\text{Δ}BYX$ .

Here,

  $\begin{array}{l}\frac{BY}{AB}=\frac{x}{12}\\ \frac{BY}{8}=\frac{x}{12}\\ BY=\frac{2x}{3}\end{array}$

Similarly, from $\text{Δ}AMN$ and $\text{Δ}ABC$ get:

  $\begin{array}{l}\frac{AM}{8}=\frac{x}{6}\\ AM=\frac{4x}{3}\end{array}$

Similarly,

  $\begin{array}{l}\frac{BR}{6}=\frac{x}{8}\\ BR=\frac{3x}{4}\end{array}$

According to the diagram MBRK is a parallelogram.

Therefore,

  $BM=8-\frac{4x}{3}$

Again,

  $\begin{array}{cc}BM& =AM-BY\\ & =\frac{4x}{3}-\frac{2x}{3}\\ & =\frac{2x}{3}\end{array}$

Compare and get:

  $\begin{array}{l}\frac{2x}{3}=8-\frac{4x}{3}\hfill \\ \frac{6x}{3}=8\hfill \\ x=4\hfill \end{array}$

Therefore, the value is $x=4$ .