   Question Answers

# In an equilateral triangle ABC, AD$\bot$ BC. Prove that $A{D^2} = 3B{D^2}$.  Hint: The altitude is also the median for an equilateral triangle. Use this property and the Pythagoras theorem (the square of the hypotenuse is the sum of the squares of the other two sides, in a right-angled triangle) to prove what is required.

ABC is an equilateral triangle. AD is perpendicular to BC.
AD$\bot$ BC. So, AD is the altitude of triangle ABC.
Let us draw the diagram. We know that, in an equilateral triangle the altitude is also the median.
Hence, D is the midpoint of BC.
BD=DC
We can also say that BD is half the length of BC.
$BD = \dfrac{1}{2}BC$ …(1)
We will use the Pythagoras theorem which states that the square of the hypotenuse is the sum of the squares of the other two sides, in a right-angled triangle.
In triangle ADB, AB is the hypotenuse.
$A{B^2} = B{D^2} + A{D^2}$
By rearranging we get,
$A{D^2} = A{B^2} - B{D^2}$ …(2)
We know that, in an equilateral triangle, all the sides are equal.
So, BC=AB …(3)
Using (3) in (1) we get,
$BD = \dfrac{1}{2}AB$
By cross-multiplying we get,
$AB = 2BD$ …(4)
Using (4) in (1) we get,
$A{D^2} = {(2BD)^2} - B{D^2} \\ A{D^2} = 4B{D^2} - B{D^2} \\ A{D^2} = 3B{D^2} \\$
Hence, $A{D^2} = 3B{D^2}$ is proved.

Note: In problems where we have to prove LHS=RHS, start from the LHS and take the proof in that direction as expressing the terms in terms of what is given in the RHS. This will lead to getting the correct RHS. Reasons must be written for each step while proving.
View Notes
Area of Equilateral Triangle Formula  Right Angled Triangle Constructions  CBSE Class 10 Maths Chapter 6 - Triangles Formula  Group Theory in Mathematics  CBSE Class 7 Maths Chapter 6 - Triangle and Its Properties Formulas  CBSE Class 10 Maths Chapter 3 - Pair of Linear Equations in Two Variables Formula  Potential Energy of Charges in an Electric Field  CBSE Class 10 Maths Chapter 10 - Circles Formula  Area of Triangle Formula  NCERT Books for Class 10 Maths PDF in Hindi  Important Questions for CBSE Class 7 English An Alien Hand Chapter 10 - An Alien Hand  Important Questions for CBSE Class 7 English An Alien Hand Chapter 9 - A Tiger In The House  Important Questions for CBSE Class 7 English An Alien Hand Chapter 6 - I want Something In A Cage  Important Questions for CBSE Class 10 Maths Chapter 6 - Triangles  Important Questions for CBSE Class 10 Maths Chapter 3 - Pair of Linear Equations in Two Variables  Important Questions for CBSE Class 6 English Honeysuckle Chapter 4 - An Indian American Woman in Space: Kalpana Chawala  Important Questions for CBSE Class 7 English An Alien Hand  Important Questions for CBSE Class 7 Science Chapter 10 - Respiration in Organisms  Important Questions for CBSE Class 10 Maths Chapter 10 - Circles  Important Questions for CBSE Class 12 Biology Chapter 10 - Microbes in Human Welfare  CBSE Class 10 Hindi A Question Paper 2020  Hindi A Class 10 CBSE Question Paper 2009  Hindi A Class 10 CBSE Question Paper 2015  CBSE Class 10 Maths Question Paper 2017  CBSE Class 10 Maths Question Paper 2020  Hindi A Class 10 CBSE Question Paper 2007  Hindi A Class 10 CBSE Question Paper 2013  Hindi A Class 10 CBSE Question Paper 2016  Hindi A Class 10 CBSE Question Paper 2012  Hindi A Class 10 CBSE Question Paper 2010  RS Aggarwal Class 9 Solutions Chapter-9 Congruence of Triangles and Inequalities in a Triangle  NCERT Solutions for Class 10 Maths Chapter 6 Triangles  NCERT Solutions for Class 10 Maths Chapter 6 Triangles in Hindi  NCERT Solutions for Class 10 Maths Chapter 10 Circles in Hindi  NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and Its Properties In Hindi  NCERT Solutions for Class 7 English An Alien Hand - A Tiger In The House  RS Aggarwal Class 10 Solutions - Triangles  NCERT Solutions for Class 7 English An Alien Hand - I Want Something In A Cage  NCERT Solutions for Class 10 Maths in Hindi  RD Sharma Solutions for Class 9 Maths Chapter 10 - Congruent Triangles  