What is the area of a quadrilateral ABCD, if AB = 5 cm, BC = 13 cm, CD = 9 cm, DA = 15 cm, AC = 12 cm?
To find the area of a quadrilateral, you can use the formula for the area of a triangle and then sum the areas of the triangles formed by the sides of the quadrilateral.
In this case, you can divide the quadrilateral ABCD into two triangles, triangle ABC and triangle CDA.
First, let's find the area of triangle ABC. We can use Heron's formula, which uses the lengths of the triangle's sides to calculate its area. Heron's formula states that the area of a triangle with side lengths a, b, and c is given by:
area = √(s(s-a)(s-b)(s-c)),
where s is the semi-perimeter of the triangle and is calculated as (a + b + c)/2.
For triangle ABC, the side lengths are AB = 5 cm, BC = 13 cm, and AC = 12 cm. Therefore, the semi-perimeter can be calculated as:
s = (AB + BC + AC)/2 = (5 + 13 + 12)/2 = 30/2 = 15 cm.
Now, we can calculate the area of triangle ABC using Heron's formula:
area_ABC = √(s(s - AB)(s - BC)(s - AC))
= √(15(15 - 5)(15 - 13)(15 - 12))
= √(15 * 10 * 2 * 3)
= √900
= 30 cm².
Next, let's find the area of triangle CDA using the same procedure.
For triangle CDA, the side lengths are CD = 9 cm, DA = 15 cm, and AC = 12 cm. Calculating the semi-perimeter:
s = (CD + DA + AC)/2 = (9 + 15 + 12)/2 = 36/2 = 18 cm.
Using Heron's formula, we can calculate the area of triangle CDA:
area_CDA = √(s(s - CD)(s - DA)(s - AC))
= √(18(18 - 9)(18 - 15)(18 - 12))
= √(18 * 9 * 3 * 6)
= √2916
= 54 cm².
Finally, we can find the total area of the quadrilateral ABCD by summing the areas of triangle ABC and triangle CDA:
area_ABCD = area_ABC + area_CDA
= 30 cm² + 54 cm²
= 84 cm².
Therefore, the area of the quadrilateral ABCD is 84 square centimeters.
since AB=5, AC=12, BC=13
ABC is a right triangle.
Similarly, ACD is a right triangle.
So, the area should be easy to figure out, eh?