A Point Oi in the Interior of a Rectangle Abcd is Joined with Each of the Vertices A, B, C and D. Prove that Ob2 + Od2 = Oc2 + Oa2 - Mathematics

Advertisements
Advertisements
Sum

A point OI in the interior of a rectangle ABCD is joined with each of the vertices A, B, C and D. Prove that  OB2 + OD2 = OC2 + OA2

Advertisements

Solution

Let ABCD be the given rectangle and let O be a point within it.
Join OA, OB, OC and OD.

Through O, draw EOF || AB. Then, ABFE is a rectangle.
In right triangles ΔOEA and ΔOFC, we have
OA2 = OE2 + AE2 and OC2 = OF2 + CF2
⇒ OA2 + OC2 = (OE2 + AE2) + (OF2 + CF2)
⇒ OA2 + OC2 = OE2 + OF2 + AE2 + CF2   ......(i)
Now, in right triangles OFB and ODE, we have
OB2 = OF2 + FB2 and OD2 = OE2 + DE2
⇒ OB2 + OD2 = (OF2 + FB2) + (OE2 + DE2)
⇒ OB2 + OD2 = OE2 + OF2 + DE2 + BF2
⇒ OB2 + OD2 = OE2 + OF2 + CF2 + AE2  [∵ DE = CF and AE = BF]....(ii)
From (i) and (ii), we get
OA2 + OC2 = OB2 + OD2.

  Is there an error in this question or solution?
Chapter 17: Pythagoras Theorem - Exercise 17.1

APPEARS IN

Frank Class 9 Maths ICSE
Chapter 17 Pythagoras Theorem
Exercise 17.1 | Q 16

RELATED QUESTIONS

If the sides of a triangle are 6 cm, 8 cm and 10 cm, respectively, then determine whether the triangle is a right angle triangle or not.


The diagonal of a rectangular field is 16 metres more than the shorter side. If the longer side is 14 metres more than the shorter side, then find the lengths of the sides of the field.


Sides of triangles are given below. Determine it is a right triangles? In case of a right triangle, write the length of its hypotenuse. 3 cm, 8 cm, 6 cm


Sides of triangle are given below. Determine it is a right triangle or not? In case of a right triangle, write the length of its hypotenuse. 13 cm, 12 cm, 5 cm


D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that AE+ BD2 = AB2 + DE2


In the given figure, ABC is a triangle in which ∠ABC> 90° and AD ⊥ CB produced. Prove that AC2 = AB2 + BC2 + 2BC.BD.


A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall at a distance a. Find the distance of the foot of the ladder from the wall.


Which of the following can be the sides of a right triangle?

1.5 cm, 2 cm, 2.5 cm

In the case of right-angled triangles, identify the right angles.


Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm.


Prove that, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the square of remaining two sides


Prove that the points A(0, −1), B(−2, 3), C(6, 7) and D(8, 3) are the vertices of a rectangle ABCD?


Identify, with reason, if the following is a Pythagorean triplet.
(3, 5, 4)


Identify, with reason, if the following is a Pythagorean triplet.
(5, 12, 13)


Identify, with reason, if the following is a Pythagorean triplet.
(10, 24, 27)


Identify, with reason, if the following is a Pythagorean triplet.
(11, 60, 61)


For finding AB and BC with the help of information given in the figure, complete following activity.

AB = BC ..........

\[\therefore \angle BAC = \]

\[ \therefore AB = BC =\] \[\times AC\]

\[ =\] \[\times \sqrt{8}\]

\[ =\] \[\times 2\sqrt{2}\]

 =


In ∆ABC, AB = 10, AC = 7, BC = 9, then find the length of the median drawn from point C to side AB.


Some question and their alternative answer are given. Select the correct alternative.

If a, b, and c are sides of a triangle and a+ b= c2, name the type of triangle.


Find the length of the hypotenuse of a right angled triangle if remaining sides are 9 cm and 12 cm.


Pranali and Prasad started walking to the East and to the North respectively, from the same point and at the same speed. After 2 hours distance between them was \[15\sqrt{2}\]

 km. Find their speed per hour.

 


In an isosceles triangle, length of the congruent sides is 13 cm and its base is 10 cm. Find the distance between the vertex opposite the base and the centroid.


In ΔMNP, ∠MNP = 90˚, seg NQ ⊥ seg MP, MQ = 9, QP = 4, find NQ.


In right angle ΔABC, if ∠B = 90°, AB = 6, BC = 8, then find AC.


In the given figure, ∠B = 90°, XY || BC, AB = 12 cm, AY = 8cm and AX : XB = 1 : 2 = AY : YC.

Find the lengths of AC and BC.


Two poles of heights 6 m and 11 m stand vertically on a plane ground. If the distance between their feet is 12 m;
find the distance between their tips.


In triangle ABC, AB = AC = x, BC = 10 cm and the area of the triangle is 60 cm2.
Find x.


In triangle ABC, given below, AB = 8 cm, BC = 6 cm and AC = 3 cm. Calculate the length of OC.



In the following figure, AD is perpendicular to BC and D divides BC in the ratio 1: 3.

Prove that : 2AC2 = 2AB2 + BC2


In figure AB = BC and AD is perpendicular to CD.
Prove that: AC2 = 2BC. DC.


ABC is a triangle, right-angled at B. M is a point on BC.
Prove that: AM2 + BC2 = AC2 + BM2.


Diagonals of rhombus ABCD intersect each other at point O.

Prove that: OA2 + OC2 = 2AD2 - `"BD"^2/2`


In the following figure, OP, OQ, and OR are drawn perpendiculars to the sides BC, CA and AB respectively of triangle ABC.

Prove that: AR2 + BP2 + CQ2 = AQ2 + CP2 + BR2


In triangle ABC, ∠B = 90o and D is the mid-point of BC.
Prove that: AC2 = AD2 + 3CD2.


If the angles of a triangle are 30°, 60°, and 90°, then shown that the side opposite to 30° is half of the hypotenuse, and the side opposite to 60° is `sqrt(3)/2` times of the hypotenuse.


Choose the correct alternative: 

In right-angled triangle PQR, if hypotenuse PR = 12 and PQ = 6, then what is the measure of ∠P? 


Find the length of diagonal of the square whose side is 8 cm.


Prove that `(sin θ + cosec θ)^2 + (cos θ + sec θ)^2 = 7 + tan^2 θ + cot^2 θ`.


Triangle XYZ is right-angled at vertex Z. Calculate the length of YZ, if XY = 13 cm and XZ = 12 cm.


Triangle PQR is right-angled at vertex R. Calculate the length of PR, if: PQ = 34 cm and QR = 33.6 cm.


The sides of a certain triangle is given below. Find, which of them is right-triangle

16 cm, 20 cm, and 12 cm


In the given figure, angle BAC = 90°, AC = 400 m, and AB = 300 m. Find the length of BC.


In the given figure, angle ACP = ∠BDP = 90°, AC = 12 m, BD = 9 m and PA= PB = 15 m. Find:
(i) CP
(ii) PD
(iii) CD


In triangle PQR, angle Q = 90°, find: PR, if PQ = 8 cm and QR = 6 cm


In triangle PQR, angle Q = 90°, find: PQ, if PR = 34 cm and QR = 30 cm


In the given figure, angle ACB = 90° = angle ACD. If AB = 10 m, BC = 6 cm and AD = 17 cm, find :
(i) AC
(ii) CD


In the given figure, AD = 13 cm, BC = 12 cm, AB = 3 cm and angle ACD = angle ABC = 90°. Find the length of DC.


A ladder, 6.5 m long, rests against a vertical wall. If the foot of the ladder is 2.5 m from the foot of the wall, find up to how much height does the ladder reach?


In the figure below, find the value of 'x'.


In the figure below, find the value of 'x'.


In the right-angled ∆LMN, ∠M = 90°. If l(LM) = 12 cm and l(LN) = 20 cm, find the length of seg MN.


The top of a ladder of length 15 m reaches a window 9 m above the ground. What is the distance between the base of the wall and that of the ladder?


Find the Pythagorean triplet from among the following set of numbers.

4, 5, 6


Find the Pythagorean triplet from among the following set of numbers.

4, 7, 8


The sides of the triangle are given below. Find out which one is the right-angled triangle?

8, 15, 17


The sides of the triangle are given below. Find out which one is the right-angled triangle?

1.5, 1.6, 1.7


The sides of the triangle are given below. Find out which one is the right-angled triangle?

40, 20, 30


From the given figure, find the length of hypotenuse AC and the perimeter of ∆ABC.


Find the length of the perpendicular of a triangle whose base is 5cm and the hypotenuse is 13cm. Also, find its area.


The foot of a ladder is 6m away from a wall and its top reaches a window 8m above the ground. If the ladder is shifted in such a way that its foot is 8m away from the wall to what height does its tip reach?


Two poles of height 9m and 14m stand on a plane ground. If the distance between their 12m, find the distance between their tops.


The length of the diagonals of rhombus are 24cm and 10cm. Find each side of the rhombus.


In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AC2 = AD2 + BC x DE + `(1)/(4)"BC"^2`


AD is perpendicular to the side BC of an equilateral ΔABC. Prove that 4AD2 = 3AB2.


In a triangle ABC right angled at C, P and Q are points of sides CA and CB respectively, which divide these sides the ratio 2 : 1.
Prove that : 9(AQ2 + BP2) = 13AB2 


In the given figure, PQ = `"RS"/(3)` = 8cm, 3ST = 4QT = 48cm.
SHow that ∠RTP = 90°.


In a right-angled triangle PQR, right-angled at Q, S and T are points on PQ and QR respectively such as PT = SR = 13 cm, QT = 5 cm and PS = TR. Find the length of PQ and PS.


∆ABC is right-angled at C. If AC = 5 cm and BC = 12 cm. find the length of AB.


A man goes 18 m due east and then 24 m due north. Find the distance of his current position from the starting point?


There are two paths that one can choose to go from Sarah’s house to James's house. One way is to take C street, and the other way requires to take B street and then A street. How much shorter is the direct path along C street?


To get from point A to point B you must avoid walking through a pond. You must walk 34 m south and 41 m east. To the nearest meter, how many meters would be saved if it were possible to make a way through the pond?


The perpendicular PS on the base QR of a ∆PQR intersects QR at S, such that QS = 3 SR. Prove that 2PQ2 = 2PR2 + QR2 


Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels at a speed of `(20 "km")/"hr"` and the second train travels at `(30 "km")/"hr"`. After 2 hours, what is the distance between them?


If in a ΔPQR, PR2 = PQ2 + QR2, then the right angle of ∆PQR is at the vertex ________


In a right angled triangle, the hypotenuse is the greatest side


Find the unknown side in the following triangles


Find the unknown side in the following triangles


An isosceles triangle has equal sides each 13 cm and a base 24 cm in length. Find its height


Find the distance between the helicopter and the ship


In triangle ABC, line I, is a perpendicular bisector of BC.
If BC = 12 cm, SM = 8 cm, find CS


The hypotenuse of a right angled triangle of sides 12 cm and 16 cm is __________


Rithika buys an LED TV which has a 25 inches screen. If its height is 7 inches, how wide is the screen? Her TV cabinet is 20 inches wide. Will the TV fit into the cabinet? Give reason


In the figure, find AR


In the given figure, ∠T and ∠B are right angles. If the length of AT, BC and AS (in centimeters) are 15, 16, and 17 respectively, then the length of TC (in centimeters) is ______.


The perimeters of two similar triangles ABC and PQR are 60 cm and 36 cm respectively. If PQ = 9 cm, then AB equals ______.


In an equilateral triangle PQR, prove that PS2 = 3(QS)2.


The perimeter of the rectangle whose length is 60 cm and a diagonal is 61 cm is ______.


In a right-angled triangle ABC, if angle B = 90°, BC = 3 cm and AC = 5 cm, then the length of side AB is ______.


In a right-angled triangle ABC, if angle B = 90°, then which of the following is true?


The longest side of a right angled triangle is called its ______.


Two angles are said to be ______, if they have equal measures.


The hypotenuse (in cm) of a right angled triangle is 6 cm more than twice the length of the shortest side. If the length of third side is 6 cm less than thrice the length of shortest side, then find the dimensions of the triangle.


The foot of a ladder is 6 m away from its wall and its top reaches a window 8 m above the ground. Find the length of the ladder.


Share
Notifications



      Forgot password?
Use app×