Name: Angle Between Two Lines
Uploaded: 2019-07-19
Duration: 6 min 13 s
Description: If two lines are perpendicular to each other, their direction vectors are also perpendicular. This means the scalar product of the direction vectors is equal...

Question

Prove that the points

(- 4, - 1), (- 2, - 4), (4, 0)

and

(2, 3)

are the vertices of a rectangle.

Accepted Answer

Let $A (- 4, - 1), B (- 2, - 4), C (4, 0)$ and $D (2, 3)$ be the given points

$A B = \sqrt{(- 2 + 4)^{2} + (- 4 + 1)^{2}} = \sqrt{13}$

$B C = \sqrt{(4 + 2)^{2} + (0 + 4)^{2}} = 2 \sqrt{13}$

$C D = \sqrt{(2 - 4)^{2} + (3 - 0)^{2}} = \sqrt{13}$

$D A = \sqrt{(2 + 4)^{2} + (3 + 1)^{2}} = 2 \sqrt{13}$

$∴ A B = C D$ and $B C = D A$

Now, we have,

$m_{1} =$ slope of $A B = \frac{- 4 + 1}{- 2 + 4} = - \frac{3}{2}$

$m_{2} =$ slope of $B C = \frac{0 + 4}{4 + 2} = \frac{4}{6} = \frac{2}{3}$

$m_{3} =$ slope of $C D = \frac{3 - 0}{2 - 4} = - \frac{3}{2}$

$m_{4} =$ slope of $A D = \frac{3 - (- 1)}{2 - (- 4)} = \frac{4}{6} = \frac{2}{3}$

Here, $m_{1} m_{2} = (- \frac{3}{2}) (\frac{2}{3}) = - 1$ , and $m_{3} m_{4} = - 1$ , and $m_{1} = m_{3}$ , and $m_{2} = m_{4}$

Therefore, we have,

$A B = C D$

$B C = D A$

Since the product of the slopes is $- 1$

$\Rightarrow A B ⊥ B C$ , and $A B ⊥ A D$ ; $C D ⊥ B C$ , and $C D ⊥ A D$

We know that two lines perpendicular to the same line are parallel to each other.

$∴$ $A B$ $∥ D C$ , and $A D ∥ B C$

Since opposite sides of a quadrilateral are parallel and equal, and each angle of it is $90 °$ .

Hence, the quadrilateral is a rectangle, and the given points are the vertices of a rectangle.

Prove that the points $(- 4, - 1), (- 2, - 4), (4, 0)$ and $(2, 3)$ are the vertices of a rectangle.

Important Questions on The Straight Lines

Learn and Prepare for any exam you want !

Prove that the points (-4,-1), (-2,-4), (4,0) and (2,3) are the vertices of a rectangle.

Important Questions on The Straight Lines

Learn and Prepare for any exam you want !

Prove that the points $(- 4, - 1), (- 2, - 4), (4, 0)$ and $(2, 3)$ are the vertices of a rectangle.