Answer

Verified

385.8k+ views

**Hint:**Here, we will draw a figure representing the given situation. We will use the fact that the diagonals of a parallelogram bisect each other at right angles. We will then find the medians for their respective triangles. Hence, this will help us to prove that $ar\left( {\vartriangle GXF} \right) = ar\left( {\vartriangle EXH} \right) = ar\left( {\vartriangle GXH} \right) = ar\left( {\vartriangle EXF} \right)$.

**Complete step-by-step answer:**We know that , in a parallelogram, the diagonals bisect each other.

This means that since, the diagonals $EG$ and $FH$ intersect each other at $X$. Hence, $X$ is the mid-point of the diagonals $EG$ and $FH$.

Now we will draw the diagram based on the given information.

We know that the diagonals bisect each other at $90^\circ $

Hence, in the triangle $EFG$, $FX$ is the median.

Now, the median of a triangle divides it in two equal triangles with equal areas.

Therefore,

$ar\left( {\vartriangle EXF} \right) = ar\left( {\vartriangle GXF} \right)$……………………$\left( 1 \right)$

Similarly,

In the triangle $FGH$, $GX$ is the median.

Therefore,

$ar\left( {\vartriangle HGX} \right) = ar\left( {\vartriangle GXF} \right)$…………….….. $\left( 2 \right)$

Also, in the triangle $EHG$, $HX$ is the median.

Therefore,

$ar\left( {\vartriangle EXH} \right) = ar\left( {\vartriangle HGX} \right)$…………………. $\left( 3 \right)$

Hence, from the equations, $\left( 1 \right)$, $\left( 2 \right)$ and $\left( 3 \right)$, we get

$ar\left( {\vartriangle EXF} \right) = ar\left( {\vartriangle GXF} \right) = ar\left( {\vartriangle EXH} \right) = ar\left( {\vartriangle HGX} \right)$

**Therefore, it is proved that $ar\left( {\vartriangle GXF} \right) = ar\left( {\vartriangle EXH} \right) = ar\left( {\vartriangle GXH} \right) = ar\left( {\vartriangle EXF} \right)$.**

**Note:**A parallelogram is a quadrilateral in which the pair of opposite sides are parallel and equal to each other. Also, each diagonal in a parallelogram divides it into two congruent triangles. The diagonals of the parallelogram bisects each other i.e. when they intersect, they are divided into two equal parts. This means that if the length of the whole diagonal is for example 6 cm then, after intersecting with another diagonal, it gets divided into two equal parts of 3 cm each. Hence, this is an important property of the parallelograms.

Recently Updated Pages

How do you evaluate cos left dfrac13pi 12 right class 10 maths CBSE

How do you rewrite the inequality left 11 2x right class 10 maths CBSE

How do you solve 4 3x 025 class 10 maths CBSE

How do you find the zeros of x3 3x2 + 6x 18 class 10 maths CBSE

Consider the following statements in respect of the class 10 maths CBSE

How do you factor 2x3 + 3x2 8x 12 class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail

Name 10 Living and Non living things class 9 biology CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths