Chapter 7.6, Problem 17WE

To determine

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

Answer to Problem 17WE

AR	RT	AT	AN	NP	AP	RN	TP
8	8	16	6	6	12	?	?

Explanation of Solution

Given information:

AR	RT	AT	AN	NP	AP	RN	TP
?	8	16	6	?	?	?	?

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 AR,

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

Therefore,

  $\begin{array}{l}AR=AT–RT\hfill \\ =16–8\hfill \\ =8\hfill \end{array}$

For NP,

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

  $\begin{array}{cc}\frac{AR}{RT}& =\frac{AN}{NP}\\ \frac{8}{8}& =\frac{6}{NP}\\ 8NP& =48\\ NP& =6\end{array}$

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,

  $\begin{array}{l}\begin{array}{l}AP=AN+NP\hfill \\ =6+6\hfill \end{array} \\ =12\end{array}$

As there are no values for the parallel lines 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
8	8	16	6	6	12	?	?