Q. 45 - Calculus

Question

Heron’s formula allows us to compute the area of a triangle from its side lengths.

(a) Explain why knowing the side lengths of a quadrilateral is not sufficient to compute the area of the quadrilateral.

(b) Explain why knowing the side lengths of a quadrilateral and the length of one diagonal is sufficient to compute the area of the quadrilateral.

(c) Explain why knowing the side lengths of a quadrilateral and the measure of the angle at one vertex is also sufficient to compute the area of the  quadrilateral.

Step-by-Step Solution

Answer

(a) The lengths of the sides of a quadrilateral do not determine the quadrilateral’s area. For example, a square with side lengths 1 unit has an area of 1 square unit, but a rhombus with side lengths 1 and opposite interior angles of 30◦ has area $\frac{\sqrt{3}}{2}$.

(b) The length of one diagonal is sufficient to compute the area of the quadrilateral because if you know the side lengths and the length of one diagonal you can decompose the quadrilateral into two triangles.

(c) The side lengths adjacent to the angle whose measure is known determine a unique triangle. You may find the length of the third side of the triangle, which is a diagonal of the quadrilateral and use the reasoning from  (b).

Step 1. Given Information

Heron’s formula allows us to compute the area of a triangle from its side lengths.

(a) Explain why knowing the side lengths of a quadrilateral is not sufficient to compute the area of the quadrilateral.

(b) Explain why knowing the side lengths of a quadrilateral and the length of one diagonal is sufficient to compute the area of the quadrilateral.

(c) Explain why knowing the side lengths of a quadrilateral and the measure of the angle at one vertex is also sufficient to compute the area of the  quadrilateral.

Part (a) Step 1. Explaining why knowing the side lengths of a quadrilateral is not sufficient to compute the area of the quadrilateral.

A quadrilateral is a polygon having only four sides.

There are different types of quadrilateral and every quadrilateral have different quality.

The lengths of the sides of a quadrilateral do not determine the  quadrilateral’s area. For example, a square with side lengths 1 unit has an  area of 1 square unit, but a rhombus with side lengths 1 and opposite interior angles of 30◦ has area $\frac{\sqrt{3}}{2}$.

That's why the side lengths of a quadrilateral is not sufficient to compute the area of the quadrilateral. 

Part (b) Step 1. Explaining why knowing the side lengths of a quadrilateral and the length of one diagonal is sufficient to compute the area of the quadrilateral.

If you know the side lengths and the length of one diagonal you can decompose the quadrilateral into two triangles.

Part (c) Step 1. Explaining why knowing the side lengths of a quadrilateral and the measure of the angle at one vertex is also sufficient to compute the area of the quadrilateral.

The side lengths adjacent to the angle whose measure is known determine a unique triangle. You may find the length of the third side of the triangle, which is a diagonal of the quadrilateral and use the reasoning from  (b).