Advertisements
Advertisements
Question
O is any point inside a rectangle ABCD.
Prove that: OB2 + OD2 = OC2 + OA2.
Advertisements
Solution
Draw rectangle ABCD with arbitrary point O within it, and then draw lines OA, OB, OC, OD. Then draw lines from point O perpendicular to the sides: OE, OF, OG, OH.
Pythagoras theorem states that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the remaining two sides.
Using Pythagorean theorem we have from the above diagram:
OA2 = AH2 + OH2 = AH2 + AE2
OC2 = CG2 + OG2 = EB2 + HD2
OB2 = EO2 + BE2 = AH2 + BE2
OD2 = HD2 + OH2 = HD2 + AE2
Adding these equalities we get:
OA2 + OC2 = AH2 + HD2 + AE2 + EB2
OB2 + OD2 = AH2 + HD2 + AE2 + EB2
From which we prove that for any point within the rectangle there is the relation
OA2 + OC2 = OB2 + OD2
Hence Proved.
APPEARS IN
RELATED QUESTIONS
ABCD is a rectangle whose three vertices are B (4, 0), C(4, 3) and D(0,3). The length of one of its diagonals is
(A) 5
(B) 4
(C) 3
(D) 25
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. 50 cm, 80 cm, 100 cm
ABC is an isosceles triangle with AC = BC. If AB2 = 2AC2, prove that ABC is a right triangle.
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?
2.5 cm, 6.5 cm, 6 cm
In the case of right-angled triangles, identify the right angles.
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)
In the given figure, M is the midpoint of QR. ∠PRQ = 90°. Prove that, PQ2 = 4PM2 – 3PR2.
Some question and their alternative answer are given. Select the correct alternative.
If a, b, and c are sides of a triangle and a2 + b2 = c2, name the type of triangle.
In the following figure, AD is perpendicular to BC and D divides BC in the ratio 1: 3.
Prove that : 2AC2 = 2AB2 + BC2
In triangle ABC, angle A = 90o, CA = AB and D is the point on AB produced.
Prove that DC2 - BD2 = 2AB.AD.
In figure AB = BC and AD is perpendicular to CD.
Prove that: AC2 = 2BC. DC.
Triangle ABC is right-angled at vertex A. Calculate the length of BC, if AB = 18 cm and AC = 24 cm.
Triangle PQR is right-angled at vertex R. Calculate the length of PR, if: PQ = 34 cm and QR = 33.6 cm.
In the given figure, angle ADB = 90°, AC = AB = 26 cm and BD = DC. If the length of AD = 24 cm; find the length of BC.
In the right-angled ∆PQR, ∠ P = 90°. If l(PQ) = 24 cm and l(PR) = 10 cm, find the length of seg QR.
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?
Each side of rhombus is 10cm. If one of its diagonals is 16cm, find the length of the other diagonals.
A man goes 18 m due east and then 24 m due north. Find the distance of his current position from the starting point?
The hypotenuse of a right angled triangle of sides 12 cm and 16 cm is __________
