Chapter 7.6, Problem 15WE

To determine

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

Answer to Problem 15WE

AR	RT	AT	AN	NP	AP	RN	TP
12	8	20	18	12	30	15	25

Explanation of Solution

Given information:

AR	RT	AT	AN	NP	AP	RN	TP
12	?	20	?	?	30	15	?

Proportional lengths: Point L and M lie on AB and CD, respectively. If NA, then AB and 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 ΔRST : PQ || RS

Then, NA

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
12	?	20	?	?	30	15	?

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,

  $\begin{array}{l}RT=AT–AR\hfill \\ =20–12\hfill \\ =8\hfill \end{array}$

For AN,

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

  $\begin{array}{cc}\frac{AR}{AT}& =\frac{AN}{AP}\\ \frac{12}{20}& =\frac{AN}{30}\\ 20AN& =360\\ AN& =18\end{array}$

For NP,

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

Therefore,

  $\begin{array}{l}\begin{array}{l}NP=AP–AN\hfill \\ =30–18\hfill \end{array}\\ =12\end{array}$

For TP,

Use the proportion,

  $\begin{array}{cc}\frac{RN}{TP}& =\frac{AN}{AP}\\ \frac{15}{TP}& =\frac{18}{30}\\ 18TP& =450\\ TP& =25\end{array}$

Hence, the new table will be shown as;

AR	RT	AT	AN	NP	AP	RN	TP
12	8	20	18	12	30	15	25