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Math  /  Geometry

QuestionWhat is the area of rhombus ABCDA B C D ?
Enter your answer in the box. Do not round at any steps. □\square units 2{ }^{2}

Studdy Solution

STEP 1

1. The vertices of the rhombus are given as points in the coordinate plane.
2. The rhombus is a quadrilateral with equal side lengths.
3. The diagonals of the rhombus intersect at right angles.

STEP 2

1. Calculate the lengths of the diagonals.
2. Use the formula for the area of a rhombus using diagonals.
3. Calculate the area.

STEP 3

Calculate the lengths of the diagonals ACAC and BDBD.
For diagonal ACAC, use the distance formula: AC=(x2−x1)2+(y2−y1)2 AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} A(−1,4),C(−1,0) A(-1, 4), C(-1, 0) AC=((−1)−(−1))2+(0−4)2 AC = \sqrt{((-1) - (-1))^2 + (0 - 4)^2} =0+(−4)2 = \sqrt{0 + (-4)^2} =16 = \sqrt{16} =4 = 4
For diagonal BDBD, use the distance formula: BD=(x2−x1)2+(y2−y1)2 BD = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} B(3,2),D(−5,2) B(3, 2), D(-5, 2) BD=(3−(−5))2+(2−2)2 BD = \sqrt{(3 - (-5))^2 + (2 - 2)^2} =(3+5)2+0 = \sqrt{(3 + 5)^2 + 0} =82 = \sqrt{8^2} =64 = \sqrt{64} =8 = 8

STEP 4

Use the formula for the area of a rhombus using diagonals: Area=12×d1×d2 \text{Area} = \frac{1}{2} \times d_1 \times d_2
Substitute the values of the diagonals: Area=12×4×8 \text{Area} = \frac{1}{2} \times 4 \times 8

STEP 5

Calculate the area:
Area=12×4×8 \text{Area} = \frac{1}{2} \times 4 \times 8 =12×32 = \frac{1}{2} \times 32 =16 = 16
The area of rhombus ABCDABCD is:
16 units2 \boxed{16} \text{ units}^2

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