Advertisements
Advertisements
Question
In ∆ABC, seg AD ⊥ seg BC, DB = 3CD.
Prove that: 2AB2 = 2AC2 + BC2
Solution
In ∆ABC, AD ⊥ BC, and BD = 3CD ...(Given)
In ∆ADC, ∠ADC = 90°
By Pythagoras' theorem,
AC2 = AD2 + CD2
2AC2 = 2AD2 - 2CD2 ...(1) [Multiplied by 2]
In ∆ADB, ∠ADB = 90°
by Pythagoras' theorem,
AB2 = AD2 + BD2
2AB2 = 2AD2 + 2BD2 ...(2) [Multiplied by 2]
Subtracting equation (1) from (2)
2AB2 - AC2 = (2AD2 + 2BD2) - (2AD2 + 2CD2)
= 2AD2 + 2BD2 - 2AD2 - 2CD2
= 2(3CD)2 - 2CD2
= 2 × 9CD2 - 2CD2 ...[Given]
= 18CD2 - 2CD2
= 16CD2
∴ 2AB2 - 2AC2 = 16CD2 ...(3)
BC = CD + DB [C-D-B]
BC = CD + 3CD = 4CD
BC2 = 16CD2 ...(4) [squaring both sides]
From (3) & (4)
2AB2 - 2AC2 = BC2
∴ 2AB2 = 2AC2 + BC2
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.
Side of a triangle is given, determine it is a right triangle.
`(2a – 1) cm, 2\sqrt { 2a } cm, and (2a + 1) cm`
ABC is a triangle right angled at C. If AB = 25 cm and AC = 7 cm, find BC.
A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall at a distance a. Find the distance of the foot of the ladder from the wall.
Prove that, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the square of remaining two sides
In a trapezium ABCD, seg AB || seg DC seg BD ⊥ seg AD, seg AC ⊥ seg BC, If AD = 15, BC = 15 and AB = 25. Find A(▢ABCD)
In triangle ABC, AB = AC and BD is perpendicular to AC.
Prove that: BD2 - CD2 = 2CD × AD
Find the length of diagonal of the square whose side is 8 cm.
Triangle XYZ is right-angled at vertex Z. Calculate the length of YZ, if XY = 13 cm and XZ = 12 cm.
Use the information given in the figure to find the length AD.
In the right-angled ∆PQR, ∠ P = 90°. If l(PQ) = 24 cm and l(PR) = 10 cm, find the length of seg QR.
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.
The foot of a ladder is 6m away from a wall and its top reaches a window 8m above the ground. If the ladder is shifted in such a way that its foot is 8m away from the wall to what height does its tip reach?
In a triangle ABC, AC > AB, D is the midpoint BC, and AE ⊥ BC. Prove that: AB2 + AC2 = 2(AD2 + CD2)
Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels at a speed of `(20 "km")/"hr"` and the second train travels at `(30 "km")/"hr"`. After 2 hours, what is the distance between them?
Find the unknown side in the following triangles
Two squares are congruent, if they have same ______.
Height of a pole is 8 m. Find the length of rope tied with its top from a point on the ground at a distance of 6 m from its bottom.
Points A and B are on the opposite edges of a pond as shown in figure. To find the distance between the two points, the surveyor makes a right-angled triangle as shown. Find the distance AB.
