in an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

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In an equilateral triangle , prove that three times the square of one side is equal to four time the square of one of its altitudes.

Click here👆to get an answer to your question ✍️ In an equilateral triangle , prove that three times the square of one side is equal to four time the square of one of its altitudes.

Question

In an equilateral triangle , prove that three times the square of one side is equal to four time the square of one of its altitudes.

Easy Open in App Solution Verified by Toppr

Given:- An equilateral triangle with one of its altitude.

Let a be the side of the equilateral triangle.

∴BE=EC= 2 BC ​ = 2 a ​ To prove:- 4AE 2 =3a 2

In △ABE, by pythagoras theorem

AB 2 =AE 2 +BE 2 a 2 =AE 2 +( 2 a ​ ) 2 ⇒AE 2 =a 2 − 4 a 2 ​ ⇒AE 2 = 4 4a 2 −a 2 ​ ⇒AE 2 = 4 3a 2 ​ ⇒4AE 2 =3a 2

Hence proved that three times the square of one side is equal to four times the square of one of its altitudes.

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In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitude.

In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitude. . Ans: Hint: Here, we will first assume that the length of the sides of the equilateral triangle is equal to ‘a’ units...

In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitude.

Last updated date: 15th Mar 2023

• Total views: 294.3k • Views today: 4.74k Answer Verified 294.3k+ views 2 likes

Hint: Here, we will first assume that the length of the sides of the equilateral triangle is equal to ‘a’ units. Then, we will drop an altitude from a vertex to one of its sides and find its length to prove the given statement.

Complete Step-by-Step solution:

Consider a triangle ABC which is equilateral.

Let us assume that the length of each side of the equilateral triangle is ‘a’ units.

Drop a perpendicular AD from the A on the side BC of the triangle.

Since AD is perpendicular to BC, we can say that

∠ADB= 90 0 ∠ADB=900 .

Also, as ABC is an equilateral triangle, so all the sides are of equal length and all the angles are equal to 60 degrees.

Therefore, AB = a units and

∠ABC= 60 0 ∠ABC=600 .

Now, according to the angle sum property of the triangle, the sum of all the angles of a triangle is equal to 180 degrees.

So, in triangle ABD, we can write:

∠ABD+∠ADB+∠BAD= 180 0 ⇒ 60 0 + 90 0 +∠BAD= 180 0 ⇒ 150 0 +∠BAD= 180 0 ⇒∠BAD= 180 0 − 150 0 = 30 0

∠ABD+∠ADB+∠BAD=1800⇒600+900+∠BAD=1800⇒1500+∠BAD=1800⇒∠BAD=1800−1500=300

We know that for any angle

θ θ , we have: cosθ= length of base

length of hypotenuse

cos⁡θ=length of baselength of hypotenuse

So, in triangle ABD, we can write:

cos(∠BAD)= AD AB ⇒cos 30 0 = AD AB ⇒ 3 – √ 2 = AD a ⇒ 3 – √ a=2×AD

cos⁡(∠BAD)=ADAB⇒cos⁡300=ADAB⇒32=ADa⇒3a=2×AD

On, squaring both sides, we get:

( 3 – √ a) 2 = (2×AD) 2 ⇒3× a 2 =4×A D 2

(3a)2=(2×AD)2⇒3×a2=4×AD2

Here, we can see that a is the length of one of the sides of the equilateral triangle and AD is the length of the altitude. So, we can write as:

3×

(length of one side)

2 =4×

(length of altitude)

2

3×(length of one side)2=4×(length of altitude)2

Hence, we have proved that three times the square of one side in an equilateral triangle is equal to four times the square of one of its altitudes.

Note: Students should note here that in an equilateral triangle, the measure of each of the angles is 60 degrees. Students should also know about the angle sum property of a triangle in order to find the value of an unknown angle. Here, one can also use other trigonometric ratios instead of cosine to arrive at the required conclusion.

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Ex 6.5, 16

Ex 6.5, 16 In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes. Given:- Equilateral triangle ABC with each side a & AD as one of its altitudes To Prove :- 3 × Square of one side = 4 × square o

Ex 6.5, 16 - Chapter 6 Class 10 Triangles (Term 1)

Last updated at March 16, 2023 by Teachoo

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Transcript

Ex 6.5, 16 In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes. Given:- Equilateral triangle ABC with each side a & AD as one of its altitudes To Prove :- 3 × Square of one side = 4 × square of one of it’s altitude ⇒ 3a2 = 4AD2 Proof:- In Δ ADB & Δ ADC AB = AC AD = AD ∠ ADB=∠ ADC Hence ∆ ADB ≅ ∆ ADC Hence , BD = DC BD = DC = 1/2BC BD = DC = 𝑎/2 Now, ADB is a right angle triangle Using Pythagoras theorem (Hypotenuse)2 = (Height)2 + (Base)2 (AB)2 = (AD)2 + (BD)2 (a)2 = AD2 + (1/2 𝑎)2 a2 = AD2 + 𝑎2/4 a2 – 𝑎2/4=𝐴𝐷2 (4𝑎2 − 𝑎2)/4=𝐴𝐷2 3𝑎2/4=𝐴𝐷2 3a2 = 4 ×𝐴𝐷2 3a2 = 4AD2 Hence proved

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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