# In an equilateral triangle ABC, AD$ \bot $ BC. Prove that $A{D^2} = 3B{D^2}$.

Last updated date: 15th Mar 2023

•

Total views: 306k

•

Views today: 5.85k

Answer

Verified

306k+ views

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.

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.

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE