Question

The quadrilateral with vertices P $(-3,2)$, Q$(-5,-5)$, R$(2,-3)$ and S $(4,4)$ is a

• A. rectangle
• B. square
• C. rhombus
• D. kite

Answer 1

The correct option is C rhombus

Sides PQ = $PQ=\sqrt{(-5+3{)}^2+(-5-2{)}^2}$
$=\sqrt{(-2{)}^2+(-7{)}^2}=\sqrt{4+49}=\sqrt{53}$
QR $=\sqrt{(2+5{)}^2+(-3+5{)}^2}$
$=\sqrt{7^2+2^2}=\sqrt{49+4}=\sqrt{53}$
PS $=\sqrt{(4+3{)}^2+(4-2{)}^2}$
$=\sqrt{7^2+2^2}=\sqrt{49+4}=\sqrt{53}$
Diagonals PR = $\sqrt{(2+3{)}^2+(-3-2{)}^2}$
= $\sqrt{5^2+5^2}=\sqrt{25+25}=5\sqrt{2}$
QS $=\sqrt{(4+5{)}^2+(4+5{)}^2}$
$=\sqrt{9^2+9^2}=\sqrt{81+81}=9\sqrt{2}$
This shows, sides PQ = QR = RS = PS and diagonals PR $\ne$ QS
$\therefore$ The quadrilateral is a rhombus.