Advertisements
Advertisements
Question
AD is perpendicular to the side BC of an equilateral ΔABC. Prove that 4AD2 = 3AB2.
Advertisements
Solution
In equilateral triangle AD ⊥ BC.
⇒ BD = DC = `"BC"/(2)` ...(In equilateral triangle altitude bisects the opposite side)
In right triangle ABD,
AB2 = AD2 + BD2
= `"AD"^2 + ("BC"/2)^2`
= `(4"AD"^2 + "BC"^2)/(4)`
= `(4"AD"^2 + "BC"^2)/(4)` ...(Since AB = BC)
⇒ 4AB2 = 4AD2 + AB2
⇒ 3AB2 = 4AD2
Hence proved..
APPEARS IN
RELATED QUESTIONS
Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals
Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.
Identify, with reason, if the following is a Pythagorean triplet.
(11, 60, 61)
In the following Figure ∠ACB= 90° and CD ⊥ AB, prove that CD2 = BD × AD
Prove that (1 + cot A - cosec A ) (1 + tan A + sec A) = 2
In the right-angled ∆PQR, ∠ P = 90°. If l(PQ) = 24 cm and l(PR) = 10 cm, find the length of seg QR.
Find the Pythagorean triplet from among the following set of numbers.
4, 7, 8
The sides of the triangle are given below. Find out which one is the right-angled triangle?
11, 12, 15
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.
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 = 2AD2 + `(1)/(2)"BC"^2`
Determine whether the triangle whose lengths of sides are 3 cm, 4 cm, 5 cm is a right-angled triangle.
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?
An isosceles triangle has equal sides each 13 cm and a base 24 cm in length. Find its height
For going to a city B from city A, there is a route via city C such that AC ⊥ CB, AC = 2x km and CB = 2(x + 7) km. It is proposed to construct a 26 km highway which directly connects the two cities A and B. Find how much distance will be saved in reaching city B from city A after the construction of the highway.
In an equilateral triangle PQR, prove that PS2 = 3(QS)2.
Two circles having same circumference are congruent.
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.
Points A and B are on the opposite edges of a pond as shown in the following figure. To find the distance between the two points, the surveyor makes a right-angled triangle as shown. Find the distance AB.
