Chapter 8.3, Problem 19PSB

a

To determine

To find: Whether $\Delta PQT$ is similar to $\Delta PRS$ or not.

Explanation of Solution

Given information : Here, we have given $\Delta PRS$ and $\Delta PQT$

Proof: Here, $\frac{PQ}{PR}=\frac{4}{10}=\frac{2}{5}$ (As, $PR=4+6=10$ )

Similarly, $\frac{PT}{PS}=\frac{6}{15}=\frac{2}{5}$ (As, $PS=6+9=15$ )

Now, In $\Delta PQT$ and $\Delta PRS$ ,

  $\frac{PQ}{PR}\cong \frac{PT}{PS}$ (Proved above)

  $\angle P\cong \angle P$ (Common)

  $\Rightarrow \Delta PQT\sim \Delta PRS$ (By SAS Congruency Rule)

b

To determine

To find: Whether $\overline{QT}$ is parallel to $\overline{RS}$ or not.

Explanation of Solution

Given information : Here, we have given $\Delta PRS$ and $\Delta PQT$

Proof: Here, $\overline{PQ}=4$ and $\overline{QR}=6$ ,

  $\Rightarrow \overline{PQ}\ne \overline{QR}$

  $\Rightarrow$ Q is not the midpoint of $\overline{PR}$ .

Similarly, $\overline{PT}=6$ and $\overline{TS}=9$ ,

  $\Rightarrow \overline{PT}\ne \overline{TS}$

  $\Rightarrow$ T is not the midpoint of $\overline{PS}$ .

According to the midpoint theorem, if two points on the any two sides of the triangle are midpoints, then the line formed by joining the midpoints is parallel to the third side of the triangle.

As, here Q and T both are not the midpoints,

  $\therefore \overline{QT}$ is not parallel to $\overline{RS}$ .