I need help with 9a Starting with the area formula A = 1/2 xy sin theta for a tr

Question

I need help with 9a

Starting with the area formula A = 1/2 xy sin theta for a triangle in terms of two sides and the angle between them, use the accompanying gure to derive the formula A = 1/4 (ad sin A + ab sin B) + bc sin C + cd sin D) for the area of a quadrilateral. Show that if A represents the area of the quadrilateral in part (a), then A (a + c)(b + d)/4, with equality holding if and only if the quadrilateral is a rectangle. Thus, the ancient rule for the area of a quadrilateral overestimates the areas of all quadrilaterals that are not rectangles, so that tax assessors could well have continued to use this convenient formula long after they came to suspect that it never led them to underestimate the areas of quadrilateral elds.

Explanation / Answer

we know A=1/2xysin (theta)

For finding the area of quadrilateral(it is in 2D)

we have to find the area of all triangles and sum it up:

area of triangle= area of (ABD +BDC)

or area of quadrilateral= area of (ABC+CDA)

Adding both we get :

2* Area= 1/2*(ad sinA+cd sinD+ab sinB+bc sinC)

so area=1/4*(ad sinA+cd sinD+ab sinB+bc sinC)

HENCE PROOVED

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.