Question

What is the area of rhombus ABCD ? Enter your answer in the box. Do not round at any steps. units² Rhombus A B C D on a coordinate plane with vertex A at negative 1 comma 0, vertex B at 5 comma negative 3, vertex C at negative 1 comma negative 6, and vertex D at negative 7 comma negative 3. Point P is at negative 4.6 comma 1.8. Dashed segments join point P to point D and point P to point A, forming right angle D P A.

Answer 1

Formula needed:

Area of rhombus = 1/2 (diagonal 1 x diagonal 2)

$\text {Length} = \sqrt{(Y_2 - Y1)^2 + (X_2-X_1)^2}$

Find the length of diagonal 1:

Diagonal 1 = AC

$\text {AC} = \sqrt{(0-(-6))^2 + (-1-(-1))^2}$

$\text {AC} = \sqrt{36}$

$\text {AC} =6 \text{ units}$

Find the length of diagonal 2:

Diagonal 2 = BD

$\text {BD} = \sqrt{(-3-(-3))^2 + (5-(-7))^2}$

$\text {BD} = \sqrt{144}$

$\text {BD} =12 \text{ units}$

Find the area:

Area = 1/2 (diagonal 1 x diagonal 2)

Area = 1/2 (6 x 12)

Area = 36 units²

Answer: The area is 36 units²

Answer 2

Given:

Rhombus ABCD

A ( -1, 0 )

B ( 5, -3 )

C ( 1, -6 )

D ( -7, -3 )

P ( -4.6, 1.8 )

P to D and P to A forming a right angle D

To find:

The area of the rhombus.

Solution:

According to the formula,

Area = 1/2 ( diagonal AC x diagonal BD )

We need the lengths of AC and BD.

Length = √(( y2 - y1 )^2 - ( c2 - c1 )^2)

To find the length of  the diagonal AC,

Diagonal 1 = AC

A ( -1, 0 ) ( x1, y1 )

C ( 1, -6 ) ( x2, y2 )

Length = √( ( -6 - 0 )^2 - ( 1 - ( -1 ) ) )

Length of AC = √36

Length of the diagonal AC = 6 units.

Find the length of diagonal BD,

B ( 5, -3 ) ( x1, y1 )

D ( -7, -3 ) ( x2, y2 )

Length = √( ( -3 - ( -3 ) )^2 - ( -7 -5 )^2

Length of BD = √144

Length of the diagonal BD = 12 units.

Hence,

Area = 1/2 (6 x 12)

Area of the Rhombus = 36 sq. units