Find the area of a rhombus if its vertices are , , and taken in order. [Hint : Area of a rhombus = (product of its diagonals)]

Understand the Question Let the given vertices taken in order be , , , and . The diagonals of the rhombus are the line segments connecting opposite vertices: and . The area of a rhombus is given by: We first calculate the lengths of diagonals and using the distance formula: Step 1 · Find Length of Diagonal Let the vertices be , , , and . Using the distance formula for and : Step 2 · Find Length of Diagonal Using the distance formula for and : Step 3 · Calculate Area of the Rhombus Using the formula : Answer > Common Mistakes Choosing Adjacent Sides Instead of Diagonals: Diagonals connect opposite vertices ( and ), not adjacent ones ( or ). Sign Errors in Distance Formula: Carefully handle negative coordinates inside parentheses, e.g., , not . Simplifying Product of Radicals: Remember that , so .

Coordinate Geometry | Exercise 7.2

Question 10

Find the area of a rhombus if its vertices are $(3, 0)$ , $(4, 5)$ , $(-1, 4)$ and $(-2, -1)$ taken in order.

[Hint : Area of a rhombus = $\dfrac{1}{2}$ (product of its diagonals)]

Solution

Understand the Question

Let the given vertices taken in order be $A(3, 0)$ , $B(4, 5)$ , $C(-1, 4)$ , and $D(-2, -1)$ .
The diagonals of the rhombus are the line segments connecting opposite vertices: $AC$ and $BD$ .
The area of a rhombus is given by: $\text{Area} = \dfrac{1}{2} \times d_1 \times d_2 = \dfrac{1}{2} \times AC \times BD$
We first calculate the lengths of diagonals $AC$ and $BD$ using the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step 1 · Find Length of Diagonal $AC$

Let the vertices be $A(3, 0)$ , $B(4, 5)$ , $C(-1, 4)$ , and $D(-2, -1)$ . Diagram 1

Using the distance formula for $A(3, 0)$ and $C(-1, 4)$ :

\begin{aligned} \text{AC} &= \sqrt{(-1 - 3)^2 + (4 - 0)^2} \\ &= \sqrt{(-4)^2 + (4)^2} \\ &= \sqrt{16 + 16} \\ &= \sqrt{32} \\ &= \sqrt{16 \times 2} \\ &= 4\sqrt{2} \text{ units} \end{aligned}

Step 2 · Find Length of Diagonal $BD$

Using the distance formula for $B(4, 5)$ and $D(-2, -1)$ :

\begin{aligned} \text{BD} &= \sqrt{(-2 - 4)^2 + (-1 - 5)^2} \\ &= \sqrt{(-6)^2 + (-6)^2} \\ &= \sqrt{36 + 36} \\ &= \sqrt{72} \\ &= \sqrt{36 \times 2} \\ &= 6\sqrt{2} \text{ units} \end{aligned}

Step 3 · Calculate Area of the Rhombus

Using the formula $\text{Area} = \dfrac{1}{2} \times d_1 \times d_2$ :

\begin{aligned} \text{Area} &= \dfrac{1}{2} \times \text{AC} \times \text{BD} \\[0.6em] &= \dfrac{1}{2} \times (4\sqrt{2}) \times (6\sqrt{2}) \\[0.6em] &= \dfrac{1}{2} \times 24 \times (\sqrt{2} \times \sqrt{2}) \\[0.6em] &= \dfrac{1}{2} \times 24 \times 2 \\[0.6em] &= 24 \text{ square units} \end{aligned}

Answer

$24\text{ square units}$

Common Mistakes

Choosing Adjacent Sides Instead of Diagonals: Diagonals connect opposite vertices ( $AC$ and $BD$ ), not adjacent ones ( $AB$ or $BC$ ).
Sign Errors in Distance Formula: Carefully handle negative coordinates inside parentheses, e.g., $(-1 - 3)^2 = (-4)^2 = +16$ , not $-16$ .
Simplifying Product of Radicals: Remember that $\sqrt{2} \times \sqrt{2} = 2$ , so $(4\sqrt{2})(6\sqrt{2}) = 24 \times 2 = 48$ .

More questions in Exercise 7.2

Find the coordinates of the point which divides the join of $(-1, 7)$ and $(4, -3)$ in the ratio $2 : 3$ .

Find the coordinates of the points of trisection of the line segment joining $(4, -1)$ and $(-2, -3)$ .

To conduct Sports Day activities, in your rectangular shaped school ground $ABCD$ , lines have been drawn with chalk powder at a distance of $1\text{ m}$ each. $100$ flower pots have been placed at a distance of $1\text{ m}$ from each other along $AD$ , as shown in Fig. 7.12. Niharika runs $\dfrac{1}{4}\text{th}$ the distance $AD$ on the 2nd line and posts a green flag. Preet runs $\dfrac{1}{5}\text{th}$ the distance $AD$ on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

Find the ratio in which the line segment joining the points $(-3, 10)$ and $(6, -8)$ is divided by $(-1, 6)$ .

Find the ratio in which the line segment joining $A(1, -5)$ and $B(-4, 5)$ is divided by the $x$ -axis. Also find the coordinates of the point of division.

If $(1, 2)$ , $(4, y)$ , $(x, 6)$ and $(3, 5)$ are the vertices of a parallelogram taken in order, find $x$ and $y$ .

Find the coordinates of a point $\text{A}$ , where $\text{AB}$ is the diameter of a circle whose centre is $(2, -3)$ and $\text{B}$ is $(1, 4)$ .

If A and B are $(-2, -2)$ and $(2, -4)$ , respectively, find the coordinates of P such that $\text{AP} = \dfrac{3}{7} \text{AB}$ and P lies on the line segment AB.

Find the coordinates of the points which divide the line segment joining $A(-2, 2)$ and $B(2, 8)$ into four equal parts.

Q10

Find the area of a rhombus if its vertices are $(3, 0)$ , $(4, 5)$ , $(-1, 4)$ and $(-2, -1)$ taken in order.

[Hint : Area of a rhombus = $\dfrac{1}{2}$ (product of its diagonals)]