Question

In a quadrilateral ABCD angle A is equal to angle D is equal to 90 degree. Prove that AC square + BD square is equal to AD square + BC square + 2 CD multiply AB.

Please tell the right answer.... ​

Answer 1

$\large\underline{\sf{Given- }}$

A quadrilateral ABCD such that

∠ A = 90°

∠ D = 90°

$\large\underline{\sf{To\:prove - }}$

AC² + BD² = AD² + BC² + 2 × CD × AB

$\large\underline{\sf{Solution-}}$

In right ∆ ABD

Using Pythagoras Theorem, we have

BD² = AB² + AD² -----(1)

In right ∆ ACD

Using Pythagoras Theorem, we have

AC² = AD² + CD² -------(2)

Construction :- Complete rectangle AECD by producing AB to E, such that AE = CD and EC = AD.

Now,

In right ∆ ACE

⟼ AC² = CE² + AE²

⟼ AC² = CE² + ( AB + BE )²

⟼ AC² = CE² + BE² + AB² + 2 × AB × BE

$\red{\bigg \{ \because \: {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy \bigg \}}$

⟼ AC² = BC² + AB² + 2 × AB × ( AE - AB )

$\: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \: AB + BE = AE\bigg \}}$

⟼ AC² = BC² + AB² + 2 × AB × AE - 2AB²

⟼ AC² = BC² - AB² + 2 × AB × CD

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \:AE \: = \: CD \bigg \}}$

⟼ AC² = BC² - ( BD² - AD² ) + 2 × AB × CD

$\red{\bigg \{ \because \:BD^{2} = AB^{2} + AD^{2}\bigg \}}$

⟼ AC² = BC² - BD² + AD² + 2 × AB × CD

⟼ AC² + BD² = AD² + BC² + 2 × CD × AB

Hence, Proved

Additional Information :-

1. Pythagoras Theorem :-

This theorem states that : In a right-angled triangle, the square of the longest side is equal to sum of the squares of remaining sides.

2. Converse of Pythagoras Theorem :-

This theorem states that : If the square of the longest side is equal to sum of the squares of remaining two sides, angle opposite to longest side is right angle.

3. Area Ratio Theorem :-

This theorem states that :- The ratio of the area of two similar triangles is equal to the ratio of the squares of corresponding sides.

4. Basic Proportionality Theorem,

If a line is drawn parallel to one side of a triangle, intersects the other two lines in distinct points, then the other two sides are divided in the same ratio.