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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 =(1)/(2) (product of its diagonals)]

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)]

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Area of quadrilaterals and triangles: word problems

Full solution

Q. 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)]

Calculate Diagonal Lengths: To find the area of the rhombus, we need to calculate the lengths of its diagonals. The diagonals of a rhombus bisect each other at right angles. The vertices given are in order, so we can find the lengths of the diagonals by calculating the distance between opposite vertices.
Find Length of Diagonal $1$ : First, let's find the length of one diagonal that connects vertices $(3,0)$ and $(-1,4)$ . We use the distance formula: $d = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}$ . $\newline$ So, $d_1 = \sqrt{((-1 - 3)^2 + (4 - 0)^2)} = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32}$ .
Find Length of Diagonal $2$ : Now, let's find the length of the other diagonal that connects vertices $(4,5)$ and $(-2,-1)$ . Again, we use the distance formula: $d = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}$ . $\newline$ So, $d_2 = \sqrt{((-2 - 4)^2 + (-1 - 5)^2)} = \sqrt{(-6)^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72}$ .
Calculate Area Formula: We have the lengths of both diagonals now: $d_1 = \sqrt{32}$ and $d_2 = \sqrt{72}$ . To find the area ( $A$ ) of the rhombus, we use the formula $A = \frac{1}{2} \times (\text{product of its diagonals})$ . $\newline$ So, $A = \frac{1}{2} \times (\sqrt{32} \times \sqrt{72})$ .
Simplify Square Roots: Let's simplify the square roots before multiplying. $\sqrt{32} = \sqrt{(16 \times 2)} = 4\sqrt{2}$ and $\sqrt{72} = \sqrt{(36 \times 2)} = 6\sqrt{2}$ . Now, $A = (\frac{1}{2}) \times (4\sqrt{2} \times 6\sqrt{2})$ .
Multiply Values: Multiplying the values inside the parentheses gives us $A = (\frac{1}{2}) \times (24 \times 2)$ because $\sqrt{2} \times \sqrt{2} = 2$ . So, $A = (\frac{1}{2}) \times 48$ .
Divide to Find Area: Finally, we calculate the area by dividing $48$ by $2$ . $A = \frac{48}{2} = 24$ . Therefore, the area of the rhombus is $24$ square units.

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