State whether the following statements are true or false. Justify your answer.
The points $A (–1, –2), B (4, 3), C (2, 5)$ and $D (–3, 0)$ in that order form arectangle.

Given:

The points $A (–1, –2), B (4, 3), C (2, 5)$ and $D (–3, 0)$ in that order form a

rectangle.

To do:

We have to find whether the given statement is true or false.

Solution:

The distance between the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$

The distance between $A(-1,-2)$ and $B(4,3) is,

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

$=\sqrt{5^{2}+5^{2}}$

$=\sqrt{25+25}$

$=5 \sqrt{2}$

The distance between $C(2,5)$ and $D(-3,0)$ is,

$C D =\sqrt{(-3-2)^{2}+(0-5)^{2}}$

$=\sqrt{(-5)^{2}+(-5)^{2}}$

$=\sqrt{25+25}$

$=5 \sqrt{2}$

The distance between $A(-1,-2)$ and $D(-3,0)$ is,

$A D=\sqrt{(-3+1)^{2}+(0+2)^{2}}$

$=\sqrt{(-2)^{2}+2^{2}}$

$=\sqrt{4+4}$

$=2 \sqrt{2}$

The distance between $B(4,3)$ and $C(2,5)$ is,

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

$=\sqrt{2^{2}+(-2)^{2}}$

$=\sqrt{4+4}$

$=2 \sqrt{2}$

The distance between $A(-1,-2)$ and $C(2,5)$ is,

$A C=\sqrt{(2+1)^{2}+(5+2)^{2}}$

$=\sqrt{3^{2}+7^{2}}$

$=\sqrt{9+49}$

$=\sqrt{58}$

The distance between $D(-3,0)$ and $B(4,3)$ is,

$D B=\sqrt{(4+3)^{2}+(3-0)^{2}}$

$=\sqrt{7^{2}+3^{2}}$

$=\sqrt{49+9}$

$=\sqrt{58}$

We know that, in a rectangle, opposite sides are equal and diagonals are equal to each other.

Here,

$A B=C D$ and $A D=B C$

$AC=BD$

Therefore, the points $A (–1, –2), B (4, 3), C (2, 5)$ and $D (–3, 0)$ in that order form a rectangle.

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Updated on: 10-Oct-2022

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