# 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.

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

Updated on 10-Oct-2022 13:28:28