Question

If each diagnol of a parallelogram seperate it into two triangle of equal area and then show that the quadrilateral is a parallelogram.​

Answer 1

$\huge\underline\mathbb{Ello}$

$\Large{\underline{\underline{\bf{Given:-}}}}$

A quadrilateral ABCD such that ita diagnol AC and BD are such that

ar(∆ABD) = ar(∆ABc) and ar(∆ABC) = ar(∆ACD)

$\Large{\underline{\underline{\bf{To\:Prove:-}}}}$

Quadrilateral ABCD is a parallelogram.

$\Large{\underline{\underline{\bf{Proof:-}}}}$

➥Since diagonol AC of the quadrilateral ABCD seperates it into two triangle of equal area.

Therefore,

ar (∆ABC) = ar (∆ACD).....................(I)

But, ar (∆ABC) + ar (∆ACD) = ar (quad.ABCD)

⟹ 2ar (∆ABC) = ar (quad.ABCD) ..................[ using (I) ]

⟹ ar (∆ABC) = ½ ar (quad.ABCD) .................(ii)

➥since diagonol BD of the quadrilateral ABCD seperates it into triangles of equal area

∴ar (∆ABD) = ar (∆BCD)................(iii)

But , ar (∆ABD) + ar (∆BCD) = ar (quad. ABCD)

⟹2ar (∆ABD) + ar (quad. ABCD)................[using (iii) ]

⟹ar (∆ABD) = ½ ar (quad. ABCD).................(iv)

from (ii) and (iv) , we get

ar (∆ABC) = ar (∆ABD)

➥since , ∆s ABC and ABD are on the same base AB. Therefore they must have equal corresponding altitudes

i.e. Altitude from C of ∆ABC = Altitude from D of ∆ABD

⟹DC || AB

➥similarly, we obtain AD || BC

Hence , quadrilateral ABCD is a parallelogram

$\Large{\underline{\underline{\bf{Hence\:Proved}}}}$

$\huge\underline\mathbb{Thanks}$

Answer 2

Step-by-step explanation:

$hey$

since AC diagonal

ar(tri.ABC)=ar(tri.ACD)

we know that sum of triangle is equal to quadrilateral

so

ar(tri.ABC)+ar(tri.ACD)=ar(quad.ABCD)

ar(tri.ABC)+ar(tri.ABC)=ABCD

2ar(ABC)=ABCD

ar(ABC)=1/2ABCD (eq-1 )

since BD diagonal

ar(tri.ABD)=ar(tri.BDC)

we know that sum of triangle is equal to quadrilateral

ar(ABD)+ar(BDC)=arABCD

ar(ABD)+ar(ABD)=arABCD

2ar(ABD)=arABCD

ar(ABD)=1/2arABCD (eq2)

when RHS are equal of eq1&eq2

so LHS are also equal

ar(ABC)=ar(ABD)

AB is base in both triangle

longitude are also equal

AD=BC& AD||BC

since

opposite side are parallel &diagonal are equal

so we can say that ABCD is parallelogram.