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. $\sqrt{100}$ units
• B. $\sqrt{122}$ units
• C. $\sqrt{167}$ units
• D. $\sqrt{181}$ units

Answer 1

The correct option is D

$\sqrt{181}$ 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

Hence PE = SF = 1 - (-5) = 6 and QE = RF = 4 - 1 = 3

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

$y$-coordinate of R = 7 + RF = 7 + 3 = 10

Therefore coordinates of R(5,10).

Now, length of PR = $\sqrt{(5-(-5{)}^2+(10-1{)}^2}$ = $\sqrt{{10}^2+9^2}$ = $\sqrt{181}$ units