Question
In a quadrilateral ABCD, $\angle\text{A}+\angle\text{D}=90^\circ.$ Prove that $AC^2 + BD^2 = AD^2 + BC^2.$
[Hint: Produce AB and DC to meet at E]
[Hint: Produce AB and DC to meet at E]
✓
Answer
Given: A quadrilateral ABCD in which $\angle\text{A}+\angle\text{D}=90^\circ.$
To prove: $AC^2 + BD^2 = AD^2 + BC^2$
Construction: Join AC and BD.
Produce AB and BC to meet at E.
Proof: In $\triangle\text{ADE}$
$\angle\text{BAD}+\angle\text{CDA}=90^\circ$ [Given]
$\angle\text{E}=90^\circ$ [Int. angle of a $\triangle$]
By Pythagoras theorem in $\triangle\text{ADE}$ and $\triangle\text{BCR},$
$AD^2 = AE^2 + DE^2 ......(i)$
$\Rightarrow BC^2 = BE^2 + EC^2 ......(ii)$
Adding (i) and (ii), we get
$AD^2 + BC^2 = AE^2 + EC^2 + DE^2 + BE^2 ......(iii)$
By Pythagoras theorem in $\triangle\text{ECA}$ and $\triangle\text{EBD},$
$AC^2 = AE^2 + CE^2 ......(iv)$
$\Rightarrow BD^2 = BE^2 + DE^2 ......(v)$
$\Rightarrow AD^2 + BC^2 = AE^2 + EC^2 + DE^2 + BE^2 .......(vi)$ [Adding (iv) and (v)]
$\Rightarrow AC^2 + BD^2 = AD^2 + BC^2$ [Using (iii)]
Hence, proved.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
The present age of a woman is 3 years more than three times the ages of her daughter. Three years hence, the woman's age will be 10 years more than twice the age of her daughter. Find their present ages.
Solve for x and y:
$\frac{3}{\text{x}+\text{y}}+\frac{2}{\text{x}-\text{y}}=2,$
$\frac{9}{\text{x}+\text{y}}-\frac{4}{\text{x}-\text{y}}=1$
→$\frac{3}{\text{x}+\text{y}}+\frac{2}{\text{x}-\text{y}}=2,$
$\frac{9}{\text{x}+\text{y}}-\frac{4}{\text{x}-\text{y}}=1$
On a horizontal plane there is a vertical tower with a flagpole on the top of the tower. At a point, 9 metres away from the foot of the tower, the angle of elevation of the top and bottom the flagpole are 60° and 30° respectively. Find the height of the tower and the flagpole mounted on it. $\big[\text{Take}\sqrt{3}=1.732\big]$
→If the sum of 7 terms of an A.P. is 49 and that of 17 terms is 289, find the sum of n terms.
Prove the following identities:
$\frac{\big(1+\tan^2\theta\big)\cot\theta}{\text{cosec}^2\theta}=\tan\theta$
→$\frac{\big(1+\tan^2\theta\big)\cot\theta}{\text{cosec}^2\theta}=\tan\theta$
Find the zeros of the following quadratic polynomial and verify the relationship between the zeros and the coefficients:
$3x^2 - x - 4$
→$3x^2 - x - 4$
Find the mean of each of the following frequency distributions:
|
Class interval
|
0-8
|
8-16 | 16-24 | 24-32 | 32-40 |
|
Frequency
|
5 | 6 | 4 | 3 | 2 |
Solve the following system of linear equation graphically and shade the region between the two lines and x-axis:
3x + 2y -11 = 0,
2x - 3y + 10 = 0.
The king, the jack and the 10 of spades are lost from a pack of 52 cards and a card is drawn from the remaining cards after shuffling. Find the probability of getting a:
-
Red card.
-
Black jack.
-
Red king.
-
10 of hearts.
The following table gives production yield per hectare of wheat of 100 farms of a village :
Change the distribution to a 'more than' type distribution and draw its ogive.
| Production yield | 40-45 | 45-50 | 50-55 | 55-60 | 60-65 | 65-70 |
| No. of farms | 4 | 6 | 16 | 20 | 30 | 24 |