Chapter 4.4, Problem 18PSB

a)

To determine

To prove: $QS\perp PR$

Explanation of Solution

Given information: $PS=SR,PQ=QR$

Formula used: congruency of triangles, linear pair axiom

Proof:

  $\begin{array}{l}PS=SR,PQ=QP(\text{given})\\ QS=QS(\text{Reflective}\text{property)}\\ \therefore \Delta PQS\cong \Delta RQS(\text{SSS})\\ \therefore \angle PSQ=\angle PSQ(CPCT)\\ \therefore \angle PSQ+\angle RSQ={180}^{\circ }(\text{linear pair axiom})\\ \therefore \angle PSQ={90}^{\circ }=\angle RSQ.\end{array}$

b)

To determine

To prove: area of $\Delta PQR$ .

Explanation of Solution

Given information: $RS=9,QS=40,QR=41,QS\perp PR.$

Formula used: $\text{Area of triangle}=\frac{1}{2}\times \text{Base}\times \text{height}.$

Proof:

  $\begin{array}{l}\text{Area}\text{of}\Delta PQR=\frac{1}{2}\times \text{PR}\times \text{QS}\\ =\frac{1}{2}\times 2\times \text{RS}\times \text{QS}\text{(S}\text{is}\text{midpoint}\text{of PR)}\\ \text{ =9}\times \text{40}\\ \text{=360}\text{square}\text{units}\text{.}\end{array}$

c)

To determine

To prove: to find relationship among 9,40 and 41.

Explanation of Solution

Given information: Pythagoras theorem.

Formula used: Pythagoras theorem.

Proof:

  $\begin{array}{l}\because 9^2+{40}^2=81+1600\\ =1681\\ ={41}^2\\ \text{hence 9,40,41 are pythagorean triples}.\end{array}$