Advertisements
Advertisements
Question
In the following figure, AD is perpendicular to BC and D divides BC in the ratio 1: 3.
Prove that : 2AC2 = 2AB2 + BC2
Advertisements
Solution
Here,
BD : DC = 1 : 3.
⇒ BD = `1/4"BC" and CD = 3/4`BC
AC2 = AD2 + CD2 and AB2 = AD2 + BD2
Therefore,
AC2 - AB2 = CD2 - BD2
= `( 3/4"BC" )^2 - ( 1/4 "BC" )^2 `
= `9/16 "BC"^2 - 1/16 "BC"^2`
= `1/2"BC"^2`
∴ 2AC2 - 2AB2 = BC2
2AC2 = 2AB2 + BC2
Hence proved.
APPEARS IN
RELATED QUESTIONS
In the following figure, O is a point in the interior of a triangle ABC, OD ⊥ BC, OE ⊥ AC and OF ⊥ AB. Show that
(i) OA2 + OB2 + OC2 − OD2 − OE2 − OF2 = AF2 + BD2 + CE2
(ii) AF2 + BD2 + CE2 = AE2 + CD2 + BF2
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.
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
If the angles of a triangle are 30°, 60°, and 90°, then shown that the side opposite to 30° is half of the hypotenuse, and the side opposite to 60° is `sqrt(3)/2` times of the hypotenuse.
Show that the triangle ABC is a right-angled triangle; if: AB = 9 cm, BC = 40 cm and AC = 41 cm
In the figure below, find the value of 'x'.
In the right-angled ∆PQR, ∠ P = 90°. If l(PQ) = 24 cm and l(PR) = 10 cm, find the length of seg QR.
In the right-angled ∆LMN, ∠M = 90°. If l(LM) = 12 cm and l(LN) = 20 cm, find the length of seg MN.
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?
A ladder 25m long reaches a window of a building 20m above the ground. Determine the distance of the foot of the ladder from the building.
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.
Each side of rhombus is 10cm. If one of its diagonals is 16cm, find the length of the other diagonals.
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
AD is perpendicular to the side BC of an equilateral ΔABC. Prove that 4AD2 = 3AB2.
The perpendicular PS on the base QR of a ∆PQR intersects QR at S, such that QS = 3 SR. Prove that 2PQ2 = 2PR2 + QR2
From the given figure, in ∆ABQ, if AQ = 8 cm, then AB =?
A flag pole 18 m high casts a shadow 9.6 m long. Find the distance of the top of the pole from the far end of the shadow.
In a quadrilateral ABCD, ∠A + ∠D = 90°. Prove that AC2 + BD2 = AD2 + BC2
[Hint: Produce AB and DC to meet at E.]
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 ______.
The hypotenuse (in cm) of a right angled triangle is 6 cm more than twice the length of the shortest side. If the length of third side is 6 cm less than thrice the length of shortest side, then find the dimensions of the triangle.
