Coordinate Geometry | Exercise 7.1

Question 1

Find the distance between the following pairs of points :

(i) $(2, 3), (4, 1)$

(ii) $(-5, 7), (-1, 3)$

(iii) $(a, b), (-a, -b)$

Solution

Understand the Question

The distance $d$ between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by the Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
This formula is derived directly from the Pythagoras theorem, where the distance is the hypotenuse of the right triangle formed by the horizontal difference $(x_2 - x_1)$ and vertical difference $(y_2 - y_1)$ .

(i) $(2, 3), (4, 1)$

Step 1 · Calculate Distance between $(2, 3)$ and $(4, 1)$

Let the points be $A(2, 3)$ and $B(4, 1)$ . Diagram 1

Using the distance formula

\begin{aligned} \text{Distance } AB &= \sqrt{(4 - 2)^2 + (1 - 3)^2} \\ &= \sqrt{2^2 + (-2)^2} \\ &= \sqrt{4 + 4} \\ &= \sqrt{8} \\ &= 2\sqrt{2} \text{ units} \end{aligned}

Answer

(i) $2\sqrt{2} \text{ units}$

(ii) $(-5, 7), (-1, 3)$

Step 1 · Calculate Distance between $(-5, 7)$ and $(-1, 3)$

Let the points be $P(-5, 7)$ and $Q(-1, 3)$ .Using the distance formula

\begin{aligned} \text{Distance } PQ &= \sqrt{(-1 - (-5))^2 + (3 - 7)^2} \\ &= \sqrt{(-1 + 5)^2 + (-4)^2} \\ &= \sqrt{4^2 + (-4)^2} \\ &= \sqrt{16 + 16} \\ &= \sqrt{32} \\ &= 4\sqrt{2} \text{ units} \end{aligned}

Answer

(ii) $4\sqrt{2} \text{ units}$

(iii) $(a, b), (-a, -b)$

Step 1 · Calculate Distance between $(a, b)$ and $(-a, -b)$

Let the points be $M(a, b)$ and $N(-a, -b)$ . Diagram 3

Using the distance formula

\begin{aligned} \text{Distance } MN &= \sqrt{(-a - a)^2 + (-b - b)^2} \\ &= \sqrt{(-2a)^2 + (-2b)^2} \\ &= \sqrt{4a^2 + 4b^2} \\ &= \sqrt{4(a^2 + b^2)} \\ &= 2\sqrt{a^2 + b^2} \text{ units} \end{aligned}

Answer

(iii) $2\sqrt{a^2 + b^2} \text{ units}$

Common Mistakes

Sign Errors with Negatives: When substituting negative coordinates, remember that $x_2 - x_1$ with a negative value becomes addition, e.g., $-1 - (-5) = -1 + 5 = 4$ , not $-6$ .
Squaring Negatives: The square of any real number is always positive, e.g., $(-2)^2 = +4$ and $(-2a)^2 = +4a^2$ , not $-4$ or $-4a^2$ .
Incorrect Square Root Simplification: Note that $\sqrt{4(a^2 + b^2)} = 2\sqrt{a^2 + b^2} \neq 2(a + b)$ . The square root does not distribute across addition.

More questions in Exercise 7.1