Question

Quadrilateral ABCD has a perimeter of 64cm with measurements as shown. What is the area of the quadrilateral?

Answer 1

Can you find length AC?

Answer 2

yes

Answer 3

but how do you figure out AD and CD

Answer 4

\((AB)^2 + (BC)^2 = (AC)^2\)

\(6^2 + 8^2 = (AC)^2\)

\(36 + 64 = (AC)^2 \)

\(AC = 10\)

Answer 5

Area of triangle ABC:

\(A = \dfrac{AB \times BC}{2} = \dfrac{6 \times 8}{2} = 24\)

Answer 6

Triangle ACD

\((AC)^2 + (CD)^2 = (AD)^2\)

\(10^2 + (CD)^2 = (AD)^2\)

\((AD)^2 - (CD)^2 = 100\) Eq. 1

Answer 7

Now we use the perimeter.

\(AB + BC + CD + AD = 64\)

\(6 + 8 + CD + AD = 64\)

\(CD + AD = 50\)

\(CD = 50 - AD\) Eq. 2

Answer 8

Now we use equations 1 and 2 as a system of equations to solve for AD and CD.
Since equation 2 is already solved for CD, we can substitute it into equation 1.

Answer 9

\((AD)^2 - (50 - AD)^2 = 100\)

\((AD)^2 - (2500 - 100AD + (AD)^2) = 100\)

\((AD)^2 - 2500 + 100AD - (AD)^2 = 100\)

\(100AD - 2500 = 100\)

\(100AD = 2600\)

\(AD = 26\)

Substitute into eq. 2 to find CD

\(CD = 50 - AD\)

\(CD = 50 - 26\)

\(CD = 24\)

Area of triangle ACD

\(A = \dfrac{AC \times CD}{2} = \dfrac{10 \times 24}{2} = 120\)

The total area is the sum of the areas of the two triangles, 24 and 120.

Total area = 24 + 120 = 144

Answer 10

@asfcahgcax
Did you follow the explanation?

Answer 11

yep I get it now
Thanks
It was wrong in this so I got confused http://assets.openstudy.com/updates/attachments/57a36c1be4b0abc5c3d1ec21-marc_d-1470328586786-screenshot_127.png

Answer 12

He got the correct side lengths, but with no explanation, but the area of triangle ACD is incorrect because he used the hypotenuse instead of the other leg in the formula.

Answer 13

You're welcome.