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Coordinate Geometry — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsCoordinate Geometry1 Mark
Question
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 a rectangle.
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Answer

True:ABCD will from a rectangle if:
  1. It is a parallelogram.
  2. Diagonals are equal.
For parallelogram: Diagonals bisect each othere.
i. e., Mid point of AC = Mid point of BD is
$\text{i.e,}\Big(\frac{-1+2}{2},\frac{-2+5}{2}\Big)=\Big(\frac{4-3}{2},\frac{3+0}{2}\Big)$
$\Rightarrow\Big(\frac{1}{2},\frac{3}{2}\Big)=\Big(\frac{1}{2},\frac{3}{2}\Big)$
Hence, ABCD is a parallelogram.
Now, Diagonal $\text{AC}=\sqrt{(2+1)^2+(5+1)^2}$
$\Rightarrow\text{AC}=\sqrt{9+49}$
$\Rightarrow\text{AC}=\sqrt{58}\text{ units}$
and Diagonal $\text{BD}=\sqrt{(-3-4)^2+(0-3)^2}$
$\Rightarrow\text{BD}=\sqrt{49+9}$
$\Rightarrow\text{BD}=\sqrt{58}\text{ units}$
Digonal AC = Digonal BD
Hence, ABCD is a rectangle.

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