Question

In parallelogram PQRS, P(-3,2), Q(2,7) and S(1,9) are three vertices. Then the length of diagonal PR is

• A. 12 units
• B. 9 units
• C. 15 units
• D. 25 units

Answer 1

The correct option is C 15 units

Draw PE, SF parallel to $x$-axis.

Draw QE, RF parallel to $y$-axis.

Now, in right triangles $△PEQ$ and $△SFR$,

PQ = SR

$\angle QPE$ = $\angle RSF$

$\angle PQE$ = $\angle SRF$

Therefore, PE = SF and QE = RF .....[by RHS congruency]

Hence, PE = SF = 2 - (-3) = 5 and QE = RF = 7 - 2 = 5

Now, $x$-coordinate of R = 1 + SF
= 1 + 5
= 6

$y$-coordinate of R = 9 + RF
= 9 + 5
= 14

Therefore coordinates of R(6,14).

$\because$ The distance between two points $(x_1,y_1)\text{and}(x_2,y_2)$ is $\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

$\therefore$ PR = $\sqrt{(6-(-3{)}^2+(14-2{)}^2}$
= $\sqrt{9^2+{12}^2}$
= $\sqrt{225}$
= 15 units