Question

If we plot these points $P (3, - 5), Q (5, - 1), R (2, 1), S (1, 2)$ on the coordinate plane and join them, do we get a straight line? Explain.

Accepted Answer

We shall assume that if we plot these points $P (3, - 5), Q (5, - 1), R (2, 1), S (1, 2)$ on the coordinate plane and join them, we shall get a quadrilateral. We shall calculate the area of this quadrilateral with the points $P (3, - 5), Q (5, - 1), R (2, 1), S (1, 2)$ as the vertices. If we obtain the area $0$ , then the four points will be collinear, or we shall say that the four points are on a straight line. If we get any non-zero value as the area, then the four given points are not on a straight line. They form a quadrilateral (or a triangle).

We draw diagonal $Q S$ and divide the quadrilateral into $2$ triangles $∆ P Q S$ and $△ R Q S$ .

Area of both triangles are following below:

Area of $Δ P Q S = \frac{1}{2} |x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})|$

$= \frac{1}{2} | 3 (- 1 - 2) + 5 (2 - (- 5)) + 1 (- 5 - (- 1)) |$

$= \frac{1}{2} | 3 \times (- 3) + 5 \times 7 + 1 \times (- 4)) |$

$= \frac{1}{2} | - 9 + 35 - 4 |$

$= \frac{1}{2} | 22 |$

$= \frac{22}{2}$

$= 11$ sq. units.

Area of $Δ R Q S = \frac{1}{2} |x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})|$

$= \frac{1}{2} | 2 (2 - (- 1)) + 1 (- 1 - 1) + 5 (1 - 2) |$

$= \frac{1}{2} | 2 \times 3 + 1 \times (- 2) + 5 \times (- 1) |$

$= \frac{1}{2} | 6 - 2 - 5 |$

$= \frac{1}{2} | - 1 |$

$= \frac{1}{2} \times 1$

$= 0.5$ sq. units.

Now total area of quadrilateral $P Q R S =$ sum of area of triangles

$=$ area of $Δ P Q S$ $+$ area of $△ R Q S$

$= 11 + 0.5$

$= 11.5$ sq. units.

So, the area of the quadrilateral $P Q R S$ is $11.5$ sq. units.

Hence, we see that area obtained is non-zero. So, the points $P (3, - 5), Q (5, - 1), R (2, 1)$ and $S (1, 2)$ are not on a straight line. These points form either a quadrilateral or a triangle.

Telangana Board Solutions for Chapter: Coordinate Geometry, Exercise 13: Do This

Telangana Board Mathematics Solutions for Exercise - Telangana Board Solutions for Chapter: Coordinate Geometry, Exercise 13: Do This

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