Question

class 9th chapter 8 example number 5

Answer 1

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Q5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

Solution:

4

Given,

Let ABCD be a quadrilateral in which diagonals ACandBD bisect each other at right angle at O.

To prove,

Quadrilateral ABCD is a square.

Proof,

In △AOBand△COD,

AO=CO (Diagonals bisect each other)

∠AOB=∠COD (Vertically opposite)

OB=OD (Diagonals bisect each other)

Therefore, △AOB≅△CODby SAS congruence condition.

Thus, AB=CD by CPCT. …………… (i)

also,

∠OAB=∠OCD (Alternate interior angles)

⇒ ⇒AB∥CD

Now,

In △AODand△COD,

AO=CO (Diagonals bisect each other)

∠AOD=∠COD (Vertically opposite)

OD=OD (Common)

Therefore, △AOD≅△COD (by SAS congruence condition).

Thus, AD=CD (by CPCT).  ………………… (ii)

also,

AD=BCandAD=CD

⇒AD=BC=CD=AB  ……………….(iii)

also,  ∠ADC=∠BCD (by CPCT).

and ∠ADC+∠BCD=180∘ (co-interior angles)

⇒2∠ADC=180∘

⇒∠ADC=90∘   ……………. (iv)

One of the interior angle is right angle.

Thus, from (i), (ii) , (iii) and (iv) the given quadrilateral ABCD is a square.

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Answer 2

HI Friends,

Here is your answer

Q No1. Show that the bisector of angle of a parallelogram from a rectangle

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