Chapter 3.8, Problem 17PSC

To determine

To prove : The congruency of $\overrightarrow{ST}\text{ and }\overrightarrow{TV}$ .

Explanation of Solution

Given information : The following information has been given

  $\angle R\cong \angle W={90}^0$

  $\overrightarrow{RX}\cong \overrightarrow{WX}$

  $RS=\frac{3}{7}RT$

  $WV=\frac{3}{7}TV$

Formula used : Basic principles of congruency and liner addition of lines is used

Proof : Thus, from the given information, we can say that

  $\Delta XRS\cong \Delta XWV$

Now, comparing the triangles XRT and XWT, we can say that

  $\overrightarrow{RX}\cong \overrightarrow{WX}$

  $\angle R\cong \angle W={90}^0$

XT is the common chord

Thus, we can say that by SAS property, $\Delta XRT\cong \Delta XWT$

Now, by virtue of congruency, we can say that

  $\overrightarrow{RT}=\overrightarrow{WT}$

Also, we know that $RS=\frac{3}{7}RT$ , and $WV=\frac{3}{7}TV$

Hence, we can say that

  $\overrightarrow{ST}\cong \overrightarrow{TV}$

Thus, the given congruency is proved.