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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 = $\frac{1}{2}$ (product of its diagonals)]
Solution :
Given: The vertices of a rhombus taken in order are $A(3, 0)$, $B(4, 5)$, $C(-1, 4)$, and $D(-2, -1)$.
To Find: The area of the rhombus $ABCD$.
Formula Used: The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. The area of a rhombus is given by $\text{Area} = \frac{1}{2} \times d_1 \times d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals.
Step 1: Calculate the length of diagonal $AC$ ($d_1$).
The vertices are $A(3, 0)$ and $C(-1, 4)$.
$d_1 = \sqrt{(-1 - 3)^2 + (4 - 0)^2}$ [Substituting coordinates into the distance formula]
$d_1 = \sqrt{(-4)^2 + (4)^2}$
$d_1 = \sqrt{16 + 16}$
$d_1 = \sqrt{32} = 4\sqrt{2}$ units.
Step 2: Calculate the length of diagonal $BD$ ($d_2$).
The vertices are $B(4, 5)$ and $D(-2, -1)$.
$d_2 = \sqrt{(-2 - 4)^2 + (-1 - 5)^2}$ [Substituting coordinates into the distance formula]
$d_2 = \sqrt{(-6)^2 + (-6)^2}$
$d_2 = \sqrt{36 + 36}$
$d_2 = \sqrt{72} = 6\sqrt{2}$ units.
Step 3: Calculate the area of the rhombus.
$\text{Area} = \frac{1}{2} \times d_1 \times d_2$
$\text{Area} = \frac{1}{2} \times (4\sqrt{2}) \times (6\sqrt{2})$ [Substituting the values of $d_1$ and $d_2$]
$\text{Area} = \frac{1}{2} \times 24 \times (\sqrt{2} \times \sqrt{2})$
$\text{Area} = \frac{1}{2} \times 24 \times 2$
$\text{Area} = 24$ square units.
Final Answer: The area of the rhombus is 24 square units.
More Questions from Class 10 Mathematics Coordinate geometry EXERCISE 7.2
- Q1: Find the coordinates of the point which divides the join of (–1, 7) and (4, –3) in the ratio 2 : 3.
- Q2: Find the coordinates of the points of trisection of the line segment joining (4, –1) and (–2, –3).
- Q3: To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD, as shown in Fig. 7.12. Niharika runs $\frac{1}{4}$th the distance AD on the 2nd line and posts a green flag. Preet runs $\frac{1}{5}$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?
- Q4: Find the ratio in which the line segment joining the points (– 3, 10) and (6, – 8) is divided by (– 1, 6).
- Q5: 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.
- Q6: If (1, 2), (4, $y$), ($x$, 6) and (3, 5) are the vertices of a parallelogram taken in order, find $x$ and $y$.
- Q7: Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, – 3) and B is (1, 4).
- Q8: If A and B are (– 2, – 2) and (2, – 4), respectively, find the coordinates of P such that AP = $\frac{3}{7}$ AB and P lies on the line segment AB.
- Q9: Find the coordinates of the points which divide the line segment joining A(– 2, 2) and B(2, 8) into four equal parts.
CBSE Solutions for Class 10 Mathematics Coordinate geometry
Chapters in CBSE - Class 10 Mathematics
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