Chapter 7.6, Problem 7CE

To determine

Toprove:The expression $3y$ and $5y$ can be used for the proportional length.

Explanation of Solution

Given information:

The expression $3y$ and $5y$ can be used for the proportional length.

Calculation:

Proportional lengths: Point $L$ and $M$ lie on $\overline{AB}$ and $\overline{CD},$ respectively. If $\frac{AL}{LB}=\frac{CM}{MD},$ then $\overline{AB}$ and $\overline{CD}$ are divided proportionally.

Consider a triangle $ABC,$ in which a line, parallel to the base of the triangle, intersecting the remaining sides at points $D$ and $E$ respectively as shown as in the textbook.

According to the proportionality of the third suggested method, the value of $y$ is $\frac{7}{4}$ and the left upper segment is 3 times of $y$ .

Therefore,

  $\begin{array}{cc}x& =3y\\ & =\frac{3\times 7}{4}\\ & =\frac{21}{4}\end{array}$

And for the lower left segment,

  $x=5y$

  x $=\frac{5\times 7}{4}$

  x $=\frac{35}{4}$

Now, check whether both are satisfying to not,

Left segment = Upper left segment + Lower left segment

  $=\frac{21}{4}+\frac{35}{4}$

  $=\frac{21+35}{4}$

  $=\frac{56}{4}$ .

  $=14$

That is equal to the original length of the left segment.

Hence, the expression, $3y$ and $5y$ can be used for the proportional length.