Chapter 7.6, Problem 12WE

To determine

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

Answer to Problem 12WE

AR	RT	AT	AN	NP	AP	RN	TP
6	4	10	9	6	15	9	15

Explanation of Solution

Given information:

AR	RT	AT	AN	NP	AP	RN	TP
6	4	?	9	?	?	?	15

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.

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,

For $AT$

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

Therefore,

  $\begin{array}{l}\begin{array}{cc}AT& =AR+RT\\ & =6+4\end{array}\\ =10\end{array}$

For $NP$

As the line $RN$ is parallel to the line $TP$ , therefore,

  $\begin{array}{cc}\frac{AR}{RT}& =\frac{AN}{NP}\\ \frac{6}{4}& =\frac{9}{NP}\\ 6NP& =36\\ NP& =6\end{array}$

For $AP$

The point $N$ lies on the line $AP$ , as shown in the given figure. The value of $AP$ will be the sum of

  $AN+NP$

Therefore,

  $\begin{array}{cc}AP& =AN+NP\\ & =9+6\\ & =15\end{array}$

For $RN$

Use the proportion,

  $\begin{array}{l}\frac{RN}{TP}=\frac{AN}{AP}\\ \frac{RN}{15}=\frac{9}{15}\\ RN=9\end{array}$

AR	RT	AT	AN	NP	AP	RN	TP
6	4	10	9	6	15	9	15