Advertisements
Advertisements
प्रश्न
In a rectangle ABCD,
prove that: AC2 + BD2 = AB2 + BC2 + CD2 + DA2.
Advertisements
उत्तर
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
संबंधित प्रश्न
The points A(4, 7), B(p, 3) and C(7, 3) are the vertices of a right traingle ,right-angled at B. Find the values of p.
PQR is a triangle right angled at P and M is a point on QR such that PM ⊥ QR. Show that PM2 = QM . MR
The perpendicular from A on side BC of a Δ ABC intersects BC at D such that DB = 3CD . Prove that 2AB2 = 2AC2 + BC2.
In the given figure, ABC is a triangle in which ∠ABC> 90° and AD ⊥ CB produced. Prove that AC2 = AB2 + BC2 + 2BC.BD.
Which of the following can be the sides of a right triangle?
2 cm, 2 cm, 5 cm
In the case of right-angled triangles, identify the right angles.
Identify, with reason, if the following is a Pythagorean triplet.
(11, 60, 61)
In the given figure, ∠DFE = 90°, FG ⊥ ED, If GD = 8, FG = 12, find (1) EG (2) FD and (3) EF
In triangle ABC, given below, AB = 8 cm, BC = 6 cm and AC = 3 cm. Calculate the length of OC.
Choose the correct alternative:
In right-angled triangle PQR, if hypotenuse PR = 12 and PQ = 6, then what is the measure of ∠P?
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
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.
2, 6, 7
Find the Pythagorean triplet from among the following set of numbers.
9, 40, 41
The sides of the triangle are given below. Find out which one is the right-angled triangle?
1.5, 1.6, 1.7
A man goes 10 m due east and then 24 m due north. Find the distance from the straight point.
AD is perpendicular to the side BC of an equilateral ΔABC. Prove that 4AD2 = 3AB2.
In a right-angled triangle ABC,ABC = 90°, AC = 10 cm, BC = 6 cm and BC produced to D such CD = 9 cm. Find the length of AD.
PQR is an isosceles triangle with PQ = PR = 10 cm and QR = 12 cm. Find the length of the perpendicular from P to QR.
∆ABC is right-angled at C. If AC = 5 cm and BC = 12 cm. find the length of AB.
Two trees 7 m and 4 m high stand upright on a ground. If their bases (roots) are 4 m apart, then the distance between their tops is ______.
