Advertisements
Advertisements
Question
In a rectangle ABCD,
prove that: AC2 + BD2 = AB2 + BC2 + CD2 + DA2.
Advertisements
Solution
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.
Since, ABCD is a rectangle angles A, B, C and D are rt. angles.
First, we consider the ΔACD, and applying Pythagoras theorem we get,
AC2 = DA2 + CD2 ....(i)
Similarly, we get from rt. angle triangle BDC we get,
BD2 = BC2 + CD2
= BC2 + AB2 ....[ In a rectangle, opposite sides are equal, ∴ CD = AB ] ...(ii)
Adding (i) and (ii)
AC2 + BD2 = AB2 + BC2 + CD2 + DA2
Hence proved.
APPEARS IN
RELATED QUESTIONS
Side of a triangle is given, determine it is a right triangle.
`(2a – 1) cm, 2\sqrt { 2a } cm, and (2a + 1) cm`
From a point O in the interior of a ∆ABC, perpendicular OD, OE and OF are drawn to the sides BC, CA and AB respectively. Prove
that :
`(i) AF^2 + BD^2 + CE^2 = OA^2 + OB^2 + OC^2 – OD^2 – OE^2 – OF^2`
`(ii) AF^2 + BD^2 + CE^2 = AE^2 + CD^2 + BF^2`
ABC is a right triangle right-angled at C. Let BC = a, CA = b, AB = c and let p be the length of perpendicular from C on AB, prove that
(i) cp = ab
`(ii) 1/p^2=1/a^2+1/b^2`
Two poles of heights 6 m and 11 m stand on a plane ground. If the distance between the feet of the poles is 12 m, find the distance between their tops.
Identify, with reason, if the following is a Pythagorean triplet.
(4, 9, 12)
Find the length diagonal of a rectangle whose length is 35 cm and breadth is 12 cm.
In the given figure, point T is in the interior of rectangle PQRS, Prove that, TS2 + TQ2 = TP2 + TR2 (As shown in the figure, draw seg AB || side SR and A-T-B)
Find the length of the hypotenuse of a right angled triangle if remaining sides are 9 cm and 12 cm.
In the given figure, AB//CD, AB = 7 cm, BD = 25 cm and CD = 17 cm;
find the length of side BC.
In triangle ABC, ∠B = 90o and D is the mid-point of BC.
Prove that: AC2 = AD2 + 3CD2.
In the figure below, find the value of 'x'.
Find the Pythagorean triplet from among the following set of numbers.
3, 4, 5
The sides of the triangle are given below. Find out which one is the right-angled triangle?
11, 12, 15
The sides of the triangle are given below. Find out which one is the right-angled triangle?
1.5, 1.6, 1.7
In ΔABC, AD is perpendicular to BC. Prove that: AB2 + CD2 = AC2 + BD2
From a point O in the interior of aΔABC, perpendicular OD, OE and OF are drawn to the sides BC, CA and AB respectively. Prove that: AF2 + BD2 + CE2 = OA2 + OB2 + OC2 - OD2 - OE2 - OF2
The perpendicular PS on the base QR of a ∆PQR intersects QR at S, such that QS = 3 SR. Prove that 2PQ2 = 2PR2 + QR2
In a right angled triangle, if length of hypotenuse is 25 cm and height is 7 cm, then what is the length of its base?
Foot of a 10 m long ladder leaning against a vertical wall is 6 m away from the base of the wall. Find the height of the point on the wall where the top of the ladder reaches.
Two squares are congruent, if they have same ______.
