Chapter 7.6, Problem 16WE

To determine

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

Answer to Problem 16WE

AR	RT	AT	AN	NP	AP	RN	TP
9	18	27	13	26	39	12	36

Explanation of Solution

Given information:

AR	RT	AT	AN	NP	AP	RN	TP
?	18	?	?	26	?	12	36

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 AN,

Use the proportion,

  $\begin{array}{l}\frac{RN}{TP}=\frac{AN}{AN+NP}\hfill \\ \frac{12}{36}=\frac{AN}{AN+26}\hfill \\ 24AN=312\hfill \\ AN=13\hfill \end{array}$

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,

  $\begin{array}{l}\begin{array}{l}AP=AN+NP\hfill \\ =13+26\hfill \end{array} \\ =39\end{array}$

For $AR$

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

  $\begin{array}{cc}\frac{AR}{RT}& =\frac{AN}{NP}\\ \frac{AR}{18}& =\frac{13}{26}\\ 26AR& =234\\ AR& =9\end{array}$

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,

  $\begin{array}{cc}AT& =AR+RT\\ & =9+18\\ & =27\end{array}$

Hence, the new table will be shown as;

AR	RT	AT	AN	NP	AP	RN	TP
9	18	27	13	26	39	12	36