Chapter 7.6, Problem 13WE

To determine

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

Answer to Problem 13WE

AR	RT	AT	AN	NP	AP	RN	TP
?	?	?	10	6	16	?	?

Explanation of Solution

Given information:

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

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
?	?	?	?	6	16	?	?

For AN,

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

Therefore,

  $\begin{array}{l}\begin{array}{l}AN=AP–NP\hfill \\ =16–6\hfill \end{array}\\ =10\end{array}$

As only two values of sides are given, 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
?	?	?	10	6	16	?	?