Chapter 3.2, Problem 19PSB

To determine

To prove : The congruency of $\Delta NRT\cong \Delta RNP$ is to be proved

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{PR}\cong \overrightarrow{NT}$

  $\overrightarrow{NO}\cong \overrightarrow{SR}$

  $\overrightarrow{ON}\cong \frac{1}{3}\overrightarrow{NP}$

  $\overrightarrow{SR}\cong \frac{1}{3}\overrightarrow{RT}$

Formula used : If all the three sides of any two triangles are congruent, then the triangles are also said to be congruent by SSS (side-side-side) property

Proof : We know that

  $\overrightarrow{ON}\cong \frac{1}{3}\overrightarrow{NP}$

  $\overrightarrow{SR}\cong \frac{1}{3}\overrightarrow{RT}$

Also, we know that $\overrightarrow{NO}\cong \overrightarrow{SR}$ . Now, we club all these three equations together, and we get

  $\overrightarrow{RT}\cong \overrightarrow{NP}$

Also, we know that RN is a common chord. Thus,

  $\overrightarrow{RN}\cong \overrightarrow{NR}$

And,

  $\overrightarrow{PR}\cong \overrightarrow{NT}$ is already given

Thus, by all these 3 equally congruent sides, we can say that $\Delta NRT\cong \Delta RNP$ by SSS property.

Hence, given congruency is proved