What is the area of a quadrilateral ABCD, if AB = 5 cm, BC = 13 cm, CD = 9 cm, DA = 15 cm, AC = 12 cm?

then you can use law of cosines to find a diagonal and you have two triangles.

(if you just have the 4 lengths and no angle, the shape is not uniquely determined.)

If you take the time to construct this quadrilateral and its diagonal, it can be seen that there are two triangles that each consist of a Pythagorean triad: 9, 12, 15 and 5,12, 13, meaning that both triangles are right angled.

Area of a right angled triangle is easy since the two short sides serve as the base and height. You can probably figure the rest out from here.

What is the area of quadrilateral ABCD, if AB = 5 cm, BC = 13 cm, CD = 9 cm, DA = 15 cm, AC = 12 cm?

rsm sucks

the area is 84

the answer is 84!!!

To find the area of a quadrilateral, you can use the formula for the area of a quadrilateral using side lengths and diagonals.

In this case, we have the side lengths AB = 5 cm, BC = 13 cm, CD = 9 cm, and DA = 15 cm, and the diagonal AC = 12 cm.

To calculate the area of quadrilateral ABCD, we can use the formula:

Area = 1/2 * (AC * BD)

First, we need to find the length of the diagonal BD. We can use the Law of Cosines to find the length of BD:

BD^2 = AB^2 + AD^2 - 2 * AB * AD * cos(angle BDA)

In this case, angle BDA is an opposite angle to angle BCD, which can be found using the Law of Cosines as well:

cos(angle BDA) = (AB^2 + AD^2 - BD^2) / (2 * AB * AD)

Substituting the given values:

(AB^2 + AD^2 - BD^2) / (2 * AB * AD) = (5^2 + 15^2 - BD^2) / (2 * 5 * 15)

Simplifying:

25 + 225 - BD^2 = 150

BD^2 = 250

BD = √250 = 5√10 cm

Now that we have the lengths of the diagonals AC and BD, we can calculate the area of quadrilateral ABCD:

Area = 1/2 * (AC * BD)

Substituting the values:

Area = 0.5 * (12 cm * 5√10 cm)

Area = 6 cm * 5√10 cm

Area = 30√10 cm²

Therefore, the area of quadrilateral ABCD is 30√10 square centimeters.