Question

In a parallelogram ABCD, X and Y are points on diagonal BD such that DX = BY. Prove that AXCY is a parallelogram.




EXPLAINATION TOO PLEASE.



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Answer 1

$\large{\text{\underline{Hint:-}}}$

Join $\overline{AC}$ and check whether two diagonals divide each other equally.

$\large{\text{\underline{Solution:-}}}$

In parallelogram ABCD, two diagonals meet at a point, say $P$. Then,

$\hookrightarrow \overline{BP}=\overline{DP}\text{ and }\overline{AP}=\overline{CP}$

Since five vertices $B,D,P,X,Y$are colinear,

$\hookrightarrow \overline{DP}=\overline{DX}+\overline{XP}\text{ and }\overline{BP}=\overline{BY}+\overline{YP}$

Rearranging the equation,

$\hookrightarrow \overline{XP}=\overline{DP}-\overline{DX}\text{ and }\overline{YP}=\overline{BP}-\overline{BY}$

But, we are given that

$\hookrightarrow \overline{DP}=\overline{BP}\text{ and }\overline{DX}=\overline{BY}$

Hence,

$\hookrightarrow \overline{XP}=\overline{YP}$

Now, from each diagonal of a quadrilateral AXCY,

$\hookrightarrow \overline{XP}=\overline{YP}\text{ and }\overline{AP}=\overline{CP}$

Hence, the two diagonals of a quadrilateral AXCY divide each other in half. So, the quadrilateral AXCY is a parallelogram.

It might be confusing, but you can do it!

$\large{\text{\underline{Learn more 1:-}}}$

These are definitions of the quadrilaterals.

Trapezoid

A quadrilateral with a pair of sides being parallel.

Parallelogram

A quadrilateral with two pairs of corresponding sides being parallel.

Rhombus

A parallelogram with four equal side lengths.

Rectangle

A parallelogram with a right angle.

Square

A square is a quadrilateral that is both a rhombus and a rectangle.

$\large{\text{\underline{Learn more 2:-}}}$

Each kind of quadrilateral satisfies the following conditions.

Trapezoid

None.

Parallelogram

❶The corresponding sides are equal in length.

❷The corresponding angles are equal in measure.

❸Two diagonals divide each other in half.

Rhombus

❶The two diagonals are perpendicular.

& It satisfies the condition of a parallelogram.

Rectangle

❶The two diagonals are equal in length.

& It satisfies the condition of a parallelogram.

Square

❶It satisfies the condition of both a parallelogram and a rhombus.

$\large{\text{\underline{Learn more 3:-}}}$

When a quadrilateral satisfies at least one of the conditions, the kind of quadrilateral can be determined.

Trapezoid

None.

Parallelogram

❶The corresponding sides are parallel.

❷The corresponding sides are equal in length.

❸The corresponding angles are equal in measure.

❹The diagonals divide each other in half.

❺One pair of corresponding sides are parallel and equal in length.

Rhombus

None.

Rectangle

None.

Square

None.

Answer 2

Given :-

In a parallelogram ABCD, X and Y are points on diagonal BD such that DX = BY

To Prove :-

Prove that AXCY is a parallelogram.

Solution :-

Properties of parallelogram before solving it

• The Opposite side of a parallelogram are equal
• Alternate angle of parallelogram are equal

In the parallelogram there are two triangle

△AYB & △CXD

According to the statement

DX = CY

∠CDX = ∠ABY

CD = BY

Using the SAS congurency

△CXD ≅ △AYB

CX = AY

_________________________________________________________

We have another two triangle

△AXD and △CYB

∠ADX = ∠CBY

According to the statement

DX = CY

AD = CB

Using SAS congruency

△AXD ≅ △CYB

AX = CY      

Since

AX = CY and  CX = AY

Thus it is a parallelogram