Chapter 7.6, Problem 14WE

To determine

To find: The correct values to complete the given table.

Answer to Problem 14WE

AR	RT	AT	AN	NP	AP	RN	TP
18	6	24	?	?	?	30	40

Explanation of Solution

Given information:

AR	RT	AT	AN	NP	AP	RN	TP
18	?	?	?	?	?	30	40

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.

Triangle Proportionality Theorem: if a line parallel to one side of a triangle intersects the other two sides, then it divided those sides proportionally.

If $\text{Δ}RST:\overline{PQ}\parallel \overline{RS}$

Then, $\frac{RP}{PT}=\frac{SQ}{QT}$

Now consider a triangle $ATP,$ in which a line intersects $AT$ and $AP,$ the sides of the triangle, at the

points $R$ and $N$ on the $AT$ and $AP$ respectively.

Thus,

AR	RT	AT	AN	NP	AP	RN	TP
18	?	?	?	?	?	30	40

For RT,

Use the proportion,

  $\begin{array}{cc}\frac{RN}{TP}& =\frac{AR}{AT}\\ \frac{30}{40}& =\frac{18}{AT}\\ AT& =\frac{720}{30}\\ & =24\end{array}$

For RT,

The point R lies on the line AT, as shown in the given figure. The value of RT will be the sum of AT−AR.

Therefore,

  $\begin{array}{cc}RT& =AT-AR\\ & =24-18\\ & =6\end{array}$

As no values of right side had given therefore, it is impossible to find out all other variables.

Hence, the new table will be shown as;

AR	RT	AT	AN	NP	AP	RN	TP
18	6	24	?	?	?	30	40