11.8.4. Heron's Formula

Find, to the nearest hundredth, the area of quadrilateral in which , cm, cm, cm, and cm.

• A
• B
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• D

Example

Find, to the nearest hundredth, the area of quadrilateral in which , cm, cm, cm, and cm.

Solution

The quadrilateral is composed of two triangles. The first triangle, , is a right-angled triangle with the sides , , and . The length of is 8 cm, and the length of is 6 cm. Use Pythagoras' theorem to find the length of .

According to Heron's formula, the area of a triangle whose side lengths are , , and is equal to , where is half the triangle's perimeter. The perimeter of is , so is equal to . Find the area of using Heron's formula.

The second triangle of which the quadrilateral is composed, , has the sides , , and . The length of is 10 cm, the length of is 30 cm, and the length of is 39 cm. The perimeter of is , so is equal to . Find the approximate area of using Heron's formula.

Thus, the area of the quadrilateral is approximately .

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