Advertisements
Advertisements
Question
In an isosceles triangle ABC; AB = AC and D is the point on BC produced.
Prove that: AD2 = AC2 + BD.CD.
Solution
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
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.
An aeroplane leaves an airport and flies due north at a speed of 1,000 km per hour. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1,200 km per hour. How far apart will be the two planes after `1 1/2` hours?
Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
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
Find the side and perimeter of a square whose diagonal is 10 cm.
In the figure: ∠PSQ = 90o, PQ = 10 cm, QS = 6 cm and RQ = 9 cm. Calculate the length of PR.
In figure AB = BC and AD is perpendicular to CD.
Prove that: AC2 = 2BC. DC.
O is any point inside a rectangle ABCD.
Prove that: OB2 + OD2 = OC2 + OA2.
In the following Figure ∠ACB= 90° and CD ⊥ AB, prove that CD2 = BD × AD
Prove that in a right angle triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.
In the figure below, find the value of 'x'.
Find the Pythagorean triplet from among the following set of numbers.
3, 4, 5
Find the length of the perpendicular of a triangle whose base is 5cm and the hypotenuse is 13cm. Also, find its area.
A man goes 10 m due east and then 24 m due north. Find the distance from the straight point.
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
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
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?
In a right angled triangle, the hypotenuse is the greatest side
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?
Sides AB and BE of a right triangle, right-angled at B are of lengths 16 cm and 8 cm respectively. The length of the side of largest square FDGB that can be inscribed in the triangle ABE is ______.
