Advertisements
Advertisements
प्रश्न
In an isosceles triangle ABC; AB = AC and D is the point on BC produced.
Prove that: AD2 = AC2 + BD.CD.
उत्तर
In an isosceles triangle ABC; AB = AC and
D is the point on BC produced.
Construct AE perpendicular BC.
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.
We consider the rt. angled ΔAED and applying Pythagoras theorem we get,
AD2 = AE2 + ED2
AD2 = AE2 + ( EC + CD )2 ....(i)[ ∵ ED = EC + CD ]
Similarly, in ΔAEC,
AC2 = AE2 + EC2
AE2 = AC2 - EC2 ....(ii)
Putting AE2 = AC2 - EC2 in (i), We get,
AD2 = AC2 - EC2 + ( EC + CD )2
= AC2 + CD( CD + 2EC )
AD2 = AC2 + BD.CD .....[ ∵ 2EC + CD = BD ]
Hence proved.
APPEARS IN
संबंधित प्रश्न
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. 7 cm, 24 cm, 25 cm
A guy wire attached to a vertical pole of height 18 m is 24 m long and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire will be taut?
In the given figure, ABC is a triangle in which ∠ABC> 90° and AD ⊥ CB produced. Prove that AC2 = AB2 + BC2 + 2BC.BD.
A tree is broken at a height of 5 m from the ground and its top touches the ground at a distance of 12 m from the base of the tree. Find the original height of the tree.
The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is
(A)\[7 + \sqrt{5}\]
(B) 5
(C) 10
(D) 12
Identify, with reason, if the following is a Pythagorean triplet.
(10, 24, 27)
If P and Q are the points on side CA and CB respectively of ΔABC, right angled at C, prove that (AQ2 + BP2) = (AB2 + PQ2)
Prove that (1 + cot A - cosec A ) (1 + tan A + sec A) = 2
A boy first goes 5 m due north and then 12 m due east. Find the distance between the initial and the final position of the boy.
In the figure below, find the value of 'x'.
Find the Pythagorean triplet from among the following set of numbers.
4, 5, 6
In ΔABC, AD is perpendicular to BC. Prove that: AB2 + CD2 = AC2 + BD2
In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AC2 - AB2 = 2BC x ED
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?
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
If S is a point on side PQ of a ΔPQR such that PS = QS = RS, then ______.
Two squares are congruent, if they have same ______.
Two poles of 10 m and 15 m stand upright on a plane ground. If the distance between the tops is 13 m, find the distance between their feet.
