Name: Introduction to Straight lines
Uploaded: 2019-07-19
Duration: 25 min 10 s
Description: In this video, basic concepts of Straight lines are discussed. The content in this video is helpful to solve the related problems on this topic.

Question

$P Q R S$ is a rectangle formed by joining the points $P (- 1, - 1), Q (- 1, 4), R (5, 4)$ and $S (5, - 1) .$ $A, B, C$ and $D$ are the midpoints of $P Q, Q R, R S$ and $S P$ respectively. Is the quadrilateral $A B C D$ a square, a rectangle or a rhombus? Justify your answer.

Accepted Answer

$A, B, C$ and $D$ are mid points of $P Q, Q R, R S & S P$ respectively.

Thus, the coordinates of mid-points $A, B, C, D$ are given by,

$A = (\frac{- 1 + (- 1)}{2}, \frac{- 1 + 4}{2}) = (- 1, \frac{3}{2})$

$B = (\frac{- 1 + 5}{2}, \frac{4 + 4}{2}) = (2, 4)$

$C = (\frac{5 + 5}{2}, \frac{4 + (- 1)}{2}) = (5, \frac{3}{2})$

$D = (\frac{5 + (- 1)}{2}, \frac{- 1 + (- 1)}{2}) = (2, - 1)$

Now, $A B = \sqrt{{(2 - (- 1))}^{2} + {(4 - \frac{3}{2})}^{2}}$

$= \sqrt{{(2 + 1)}^{2} + {(\frac{8 - 3}{2})}^{2}} = \sqrt{9 + \frac{25}{4}} = \sqrt{\frac{36 + 25}{4}} = \sqrt{\frac{61}{4}} = \frac{\sqrt{61}}{2}$

$B C = \sqrt{{(5 - 2)}^{2} + {(\frac{3}{2} - 4)}^{2}}$

$= \sqrt{{(3)}^{2} + {(\frac{3 - 8}{2})}^{2}} = \sqrt{9 + \frac{25}{4}} = \sqrt{\frac{36 + 25}{4}} = \sqrt{\frac{61}{4}} = \frac{\sqrt{61}}{2}$

$C D = \sqrt{{(2 - 5)}^{2} + {(- 1 - \frac{3}{2})}^{2}}$

$= \sqrt{{(- 3)}^{2} + {(\frac{- 2 - 3}{2})}^{2}} = \sqrt{9 + \frac{25}{4}} = \sqrt{\frac{36 + 25}{4}} = \sqrt{\frac{61}{4}} = \frac{\sqrt{61}}{2}$

$D A = \sqrt{{(- 1 - 2)}^{2} + {(\frac{3}{2} - (- 1))}^{2}}$

$= \sqrt{{(- 3)}^{2} + {(\frac{3}{2} + 1)}^{2}} = \sqrt{{(- 3)}^{2} + {(\frac{3 + 2}{2})}^{2}} = \sqrt{9 + \frac{25}{4}} = \sqrt{\frac{36 + 25}{4}} = \sqrt{\frac{61}{4}} = \frac{\sqrt{61}}{2}$

Thus, $A B = B C = B C = D A$ .

So, $A B C D$ can be a square or a rhombus.

But, we do not know about the measure of angles. Hence, we cannot say that this is a square.

Hence, $A B C D$ is rhombus.

$P Q R S$ is a rectangle formed by joining the points $P (- 1, - 1), Q (- 1, 4), R (5, 4)$ and $S (5, - 1) .$ $A, B, C$ and $D$ are the midpoints of $P Q, Q R, R S$ and $S P$ respectively. Is the quadrilateral $A B C D$ a square, a rectangle or a rhombus? Justify your answer.

Important Questions on Coordinate Geometry

Learn and Prepare for any exam you want !

PQRS is a rectangle formed by joining the points P(-1,-1), Q(-1,4), R(5,4) and S(5,-1) . A ,B, C and D are the midpoints of PQ, QR, RS and SP respectively. Is the quadrilateral ABCD a square, a rectangle or a rhombus? Justify your answer.

Important Questions on Coordinate Geometry

Learn and Prepare for any exam you want !

$P Q R S$ is a rectangle formed by joining the points $P (- 1, - 1), Q (- 1, 4), R (5, 4)$ and $S (5, - 1) .$ $A, B, C$ and $D$ are the midpoints of $P Q, Q R, R S$ and $S P$ respectively. Is the quadrilateral $A B C D$ a square, a rectangle or a rhombus? Justify your answer.