If each diagonal of a quadrilateral separates it into two triangles of equal area, then show that the quadrilateral is a parallelogram.

Asked by Topperlearning User | 17th Aug, 2017, 12:24: PM

Expert Answer:

ar(ABD) = ar(BDC) and ar(ACB) = ar(ADC)
ar(ABD) + ar(BDC) = ar(Quad. ABCD)
                   2ar(ABD) = ar(Quad. ABCD)
                      ar(ABD) = begin mathsize 12px style 1 half end stylear(Quad. ABCD)
      Similarly ar(ADC) = size 12px 1 over size 12px 2ar(Quad. ABCD)
                  ar(ABD) = ar(ADC)
ABD and ADC are on the same base AD
Altitude from B of the ABD = altitude from C of the ADC
AD  BC    Similarly, AB  DC
Quadrilateral ABCD is a parallelogram     (opposite sides are parallel)

Answered by  | 17th Aug, 2017, 02:24: PM