Download App

Triangles — Maths STD 10 — Question

Gujarat BoardEnglish MediumSTD 10MathsTriangles3 Marks
Question
$O$ is any point inside a rectangle $ABCD$ see Fig. Prove that $OB^2 + OD^2 = OA^2 + OC^2$
.
✓

Answer

Through $O,$ draw $PQ \| BC$ so that $P$ lies on $AB$ and $Q$ lies on $DC.$
Now, $PQ \| BC$
Therefore, $PQ \perp AB$ and $PQ \perp  DC ( \angle B = 90^\circ$ and $\angle C = 90^\circ )$
So, $\angle BPQ = 90^\circ$ and $\angle CQP = 90^\circ$
Therefore, $BPQC$ and $APQD$ are both rectangles.
Now, from $\triangle OPB$
$OB^2 = BP^2 + OP^2 ...(1)$
Similarly, from $\triangle OQD,$
$OD^2 = OQ^2 + DQ^2 ...(2)$
From $\triangle OQC,$ we have
$OC^2 = OQ^2 + CQ^{2 }...(3)$
and from $\triangle OAP,$ we have
$OA^2 = AP^2 + OP^2 ...(4)$
Adding $(1)$ and $(2),$
$OB^2 + OD^2 = BP^2 + OP^2+ OQ^2 + DQ^2$
$= CQ^2 + OP^2 + OQ^2 + AP^2 ($As $BP = CQ$ and $DQ = AP)$
$= CQ^2 + OQ^2 + OP^2 + AP^2$
$OB^2 + OD^{2 }= OC^2 + OA^2 [$From $(3)$ and $(4)]$
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.

Start Generating Free

Explore more

More "3 Marks Question" from Triangles→Triangles — all question types→Maths STD 10 question papers→Gujarat Board STD 10 question papers→Gujarat Board question paper generator→

Similar questions

The $n^{th}$ term of an A.P is given by $(-4n + 15),$ Find the sum of first $20$ terms of this A.P.
→
Evaluate the following:
$ \frac{\tan^260^\circ+4\cos^245^\circ+3\sec^230^\circ+5\cos^290^\circ}{\text{cosec30}^\circ+\sec60^\circ-\cot^230^\circ}$
→
A box contains $90$ discs which are numbered from $1$ to $90.$ If one discs is drawn at random from the box, find the probability that it bears:
  1. A two digit number.
  2. A perfect square number.
  3. A number divisible by $5.$
→
A spherical ball of radius $3\ cm$ is melted and recast into three spherical balls. The radii of two of these balls $1.5\ cm$ and $2\ cm.$ Find t radius of the third ball
→
A toy is in the form of a cone of radius $3.5\ cm$ mounted on a hemisphere of the same radius. The total height of the toy is $15.5\ cm.$ Find the total surface area of the toy.
→
If $A(1, 2) B(4, 3)$ and $C(6, 6)$ are the three vertices of a parallelogram $ABCD,$ find the coordinates of fourth vertex $D.$
→
A lawn is in the form of a rectangle whose sides are in the ratio $5 : 3.$ The area of the lawn is $3375m^2$. Find the cost of fencing the lawn at $₹\ 65$ per metre.
→
For what value of k the following system of equations will be inconsistent?
$4x + 6y = 11$
$2x + ky = 7$
→
Prove the following trigonometric identities.
$\frac{\tan\theta}{1-\cot\theta}+\frac{\cot\theta}{1-\tan\theta}=1+\tan\theta+\cot\theta$
→
The following are quadratic equations in $x?$
$\text{x}^2-\frac{1}{\text{x}^2}=5$
→