Advertisements
Advertisements
Question
In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AB2 = AD2 - BC x CE + `(1)/(4)"BC"^2`
Advertisements
Solution
We have ∠AED = 90°
∴ ∠ADE < 90° and ∠ADC > 90°
i.e. ∠ADE is acute and ∠ADC is obtuse.
In ΔABD, ∠ADE is an acute angle.
∴ AB2 = AD2 + BD2 - 2 x BD x DE
⇒ AB2 = AD2 + `(1/2"BC")^2 - 2 xx (1)/(2)"BC" xx "DE"`
⇒ AB2 = AD2 + `(1)/(4)"BC"^2 - "BC" xx "DE"`
⇒ AB2 = AD2 + BC x DE - `(1)/(4)"BC"^2`. ....(ii)
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
ABCD is a rhombus. Prove that AB2 + BC2 + CD2 + DA2= AC2 + BD2
Tick the correct answer and justify: In ΔABC, AB = `6sqrt3` cm, AC = 12 cm and BC = 6 cm.
The angle B is:
Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
In ∆ABC, seg AD ⊥ seg BC, DB = 3CD.
Prove that: 2AB2 = 2AC2 + BC2
A man goes 40 m due north and then 50 m due west. Find his distance from the starting point.
In the given figure, AB//CD, AB = 7 cm, BD = 25 cm and CD = 17 cm;
find the length of side BC.
In a quadrilateral ABCD, ∠B = 90° and ∠D = 90°.
Prove that: 2AC2 - AB2 = BC2 + CD2 + DA2
In ∆ ABC, AD ⊥ BC.
Prove that AC2 = AB2 +BC2 − 2BC x BD
The sides of the triangle are given below. Find out which one is the right-angled triangle?
40, 20, 30
From the given figure, find the length of hypotenuse AC and the perimeter of ∆ABC.
A ladder 15m long reaches a window which is 9m above the ground on one side of a street. Keeping its foot at the same point, the ladder is turned to other side of the street to reach a window 12m high. Find the width of the street.
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 = AE2 + CD2 + BF2
In a right-angled triangle PQR, right-angled at Q, S and T are points on PQ and QR respectively such as PT = SR = 13 cm, QT = 5 cm and PS = TR. Find the length of PQ and PS.
A man goes 18 m due east and then 24 m due north. Find the distance of his current position from the starting point?
If length of sides of a triangle are a, b, c and a2 + b2 = c2, then which type of triangle it is?
Two angles are said to be ______, if they have equal measures.
Jayanti takes shortest route to her home by walking diagonally across a rectangular park. The park measures 60 metres × 80 metres. How much shorter is the route across the park than the route around its edges?
The foot of a ladder is 6 m away from its wall and its top reaches a window 8 m above the ground. Find the length of the ladder.
