Question

show that points A(1,-5),B(-4,-8),C(-1,-13) and D(4,-10) are the vertices of a rhombus​

Answer 1

Answer:

Step-by-step explanation:

Given:

• Point A (1,-5)
• Point B (-4, -8)
• Point C (-1, -13)
• Point D (4, - 10)

To Prove:

• The points are vertices of a rhombus

Proof:

Here we have to prove that the points are the vertices of a rhombus.

In a rhombus we know that all sides are equal.

Therefore we have to prove that all sides are equal.

By distance formula we know that the distance between two points is given by,

$\sf Distance\:between\:two\:points=\sqrt{(x_2-x_1)^{2} -(y_2-y_1)^{2} }$

Hence finding the distance between the points,

$\sf AB=\sqrt{(-4-1)^{2} +(-8+5)^{2} }$

$\implies \sqrt{25+9}$

$\implies \sf \sqrt{34}\: units-----(1)$

$\sf BC=\sqrt{(-1+4)^{2} +(-13+8)^{2} }$

$\implies \sqrt{9+25}$

$\implies \sf \sqrt{34}\: units----(2)$

$\sf CD=\sqrt{(4+1)^{2} +(-10+13)^{2} }$

$\implies \sqrt{25+9}$

$\implies \sf \sqrt{34}\: units----(3)$

$\sf DA=\sqrt{(1-4)^{2} +(-5+10)^{2} }$

$\implies \sqrt{9+25}$

$\implies \sf \sqrt{34}\: units----(4)$

From equations 1, 2, 3, 4

AB = BC = CD = DA

ie, all the 4 sides are equal.

Therefore the the points are vertices of a rhombus.

Hence proved.

Answer 2

Answer:

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