Math / Geometry

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

Studdy Solution

STEP 1

1. The rhombus $ABCD$ is defined by the vertices $A(1, 0)$ , $B(7, -2)$ , $C(1, -4)$ , and $D(-5, -2)$ .
2. The diagonals of the rhombus intersect at right angles.
3. The diagonals can be calculated using the distance formula.

STEP 2

1. Calculate the lengths of the diagonals $AC$ and $BD$ .
2. Use the formula for the area of a rhombus using its diagonals.
3. Calculate the area.

STEP 3

Calculate the length of diagonal $AC$ using the distance formula:
The coordinates of $A$ are $(1, 0)$ and $C$ are $(1, -4)$ .
$AC = \sqrt{(1 - 1)^2 + (0 - (-4))^2}$ $= \sqrt{0 + 4^2}$ $= \sqrt{16}$ $= 4$

STEP 4

Calculate the length of diagonal $BD$ using the distance formula:
The coordinates of $B$ are $(7, -2)$ and $D$ are $(-5, -2)$ .
$BD = \sqrt{(7 - (-5))^2 + (-2 - (-2))^2}$ $= \sqrt{12^2 + 0}$ $= \sqrt{144}$ $= 12$

STEP 5

Use the formula for the area of a rhombus using its diagonals:
$\text{Area} = \frac{1}{2} \times AC \times BD$
Substitute the values:
$\text{Area} = \frac{1}{2} \times 4 \times 12$

STEP 6

Calculate the area:
$\text{Area} = \frac{1}{2} \times 4 \times 12$ $= 2 \times 12$ $= 24$
The area of rhombus $ABCD$ is:
$\boxed{24} \text{ units}^2$

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

Studdy Solution

STEP 1

1. The rhombus $ABCD$ is defined by the vertices $A(1, 0)$ , $B(7, -2)$ , $C(1, -4)$ , and $D(-5, -2)$ .
2. The diagonals of the rhombus intersect at right angles.
3. The diagonals can be calculated using the distance formula.

STEP 2

1. Calculate the lengths of the diagonals $AC$ and $BD$ .
2. Use the formula for the area of a rhombus using its diagonals.
3. Calculate the area.

STEP 3

Calculate the length of diagonal $AC$ using the distance formula:
The coordinates of $A$ are $(1, 0)$ and $C$ are $(1, -4)$ .
$AC = \sqrt{(1 - 1)^2 + (0 - (-4))^2}$ $= \sqrt{0 + 4^2}$ $= \sqrt{16}$ $= 4$

STEP 4

Calculate the length of diagonal $BD$ using the distance formula:
The coordinates of $B$ are $(7, -2)$ and $D$ are $(-5, -2)$ .
$BD = \sqrt{(7 - (-5))^2 + (-2 - (-2))^2}$ $= \sqrt{12^2 + 0}$ $= \sqrt{144}$ $= 12$

STEP 5

Use the formula for the area of a rhombus using its diagonals:
$\text{Area} = \frac{1}{2} \times AC \times BD$
Substitute the values:
$\text{Area} = \frac{1}{2} \times 4 \times 12$

STEP 6

Calculate the area:
$\text{Area} = \frac{1}{2} \times 4 \times 12$ $= 2 \times 12$ $= 24$
The area of rhombus $ABCD$ is:
$\boxed{24} \text{ units}^2$

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

Studdy Solution

STEP 1

1. The rhombus ABCDABCDABCD is defined by the vertices A(1,0)A(1, 0)A(1,0), B(7,−2)B(7, -2)B(7,−2), C(1,−4)C(1, -4)C(1,−4), and D(−5,−2)D(-5, -2)D(−5,−2).2. The diagonals of the rhombus intersect at right angles.3. The diagonals can be calculated using the distance formula.

STEP 2

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

STEP 3

STEP 4

STEP 5

Use the formula for the area of a rhombus using its diagonals:Area=12×AC×BD \text{Area} = \frac{1}{2} \times AC \times BD Area=21​×AC×BDSubstitute the values:Area=12×4×12 \text{Area} = \frac{1}{2} \times 4 \times 12 Area=21​×4×12

STEP 6

Calculate the area:Area=12×4×12 \text{Area} = \frac{1}{2} \times 4 \times 12 Area=21​×4×12 =2×12 = 2 \times 12 =2×12 =24 = 24 =24 The area of rhombus ABCDABCDABCD is:24 units2 \boxed{24} \text{ units}^2 24​ units2

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

1. The rhombus $ABCD$ is defined by the vertices $A(1, 0)$ , $B(7, -2)$ , $C(1, -4)$ , and $D(-5, -2)$ .
2. The diagonals of the rhombus intersect at right angles.
3. The diagonals can be calculated using the distance formula.

1. Calculate the lengths of the diagonals $AC$ and $BD$ .
2. Use the formula for the area of a rhombus using its diagonals.
3. Calculate the area.

Use the formula for the area of a rhombus using its diagonals:
$\text{Area} = \frac{1}{2} \times AC \times BD$
Substitute the values:
$\text{Area} = \frac{1}{2} \times 4 \times 12$

Calculate the area:
$\text{Area} = \frac{1}{2} \times 4 \times 12$ $= 2 \times 12$ $= 24$
The area of rhombus $ABCD$ is:
$\boxed{24} \text{ units}^2$