Q10 of 24 Page 252

Prove that the area of a rhombus is equal to half of the product of the diagonals.

Consider rhombus PQRS as shown with diagonals intersecting at point A


Property of rhombus diagonals intersect at 90°



From figure area(PQRS) = area(ΔPQS) + area(ΔRQS) …(i)


Consider ΔPQS


Base = SQ


Height = PA


area(ΔPQS) = × SQ × PA …(ii)


Consider ΔSQR


Base = SQ


Height = RA


area(ΔSQR) = × SQ × RA …(iii)


substitute (ii) and (iii) in (i)


⇒ area(PQRS) = × SQ × PA + × SQ × RA


= × SQ × (PA + RA)


From figure PA + RA = PR


Therefore, area(PQRS) = × SQ × PR


Hence, the area of a rhombus is equal to half of the product of the diagonals


More from this chapter

All 24 →
8

PQRS and ABRS are parallelograms and X is any point on the side BR. Show that

(i) ar(PQRS) = ar(ABRS)


(ii) ar(∆AXS) = ar(PQRS)


 9

A farmer has a field in the form of a parallelogram PQRS as shown in the figure. He took the mid- point A on RS and joined it to points P and Q. In how many parts of field is divided? What are the shapes of these parts?

The farmer wants to sow groundnuts which are equal to the sum of pulses and paddy. How should he sow? State reasons?


 1

In a triangle ABC (see figure), E is the midpoint of median AD, show that

(i) ar ∆ABE = ar∆ACE


(ii) ar ∆ABE = ar(∆ABC)


 2

Show that the diagonals of a parallelogram divide it into four triangles of equal area.