Find The Altitude Of An Equilateral Triangle Whose Side is 20 GMAT Problem Solving

Question: Find the altitude of an equilateral triangle whose side is 20.

1. 10
2. 10√2
3. 10√3
4. 20√2
5. 20√3

“Find the altitude of an equilateral triangle whose side is 20.”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “The Official Guide for GMAT Reviews”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:
It is asked in the question to find out the height of an equilateral triangle whose side is given to be 20 units.
We know that the area of a triangle = 1/2 * base * height
In the case of a equilateral triangle, all sides are of equal length
We get,
Area = 1/2 * side * height
Equilateral triangle is a special kind of triangle which has all sides same and angles are 60 degrees each.
An equilateral triangle’s area can be found as,
Area =\(\sqrt{3}\)/4 \(a^2\) where a = side of the triangle
Now we are given the side of the triangle as 20 units
Area = \(\sqrt{3}\)/4 \(a^2\) = \(\sqrt{3}\)/4 \((20)^2\) = \(\sqrt{3}\)/4 * 400 = 100\(\sqrt{3}\)
Now in general, area of triangle =1/2 * base * height
1/2* base * height = 100 \(\sqrt{3}\)
1/2* (20) * height = 100\(\sqrt{3}\)
10 * height = 100\(\sqrt{3}\)
Height = 100/10 \(\sqrt{3}\)
Height = 10\(\sqrt{3}\)

Correct Answer: C

Approach Solution 2:
It is asked in the question to find out the height of an equilateral triangle whose side is given to be 20 units.
There is a property that the altitude of an equilateral triangle creates two equal right angle triangles and divides the base into two portions of equal size.
Considering one of the right angled triangles,
The side will become the hypotenuse = 20
The altitude will be the height of the triangle = x
The base will be the half of the length of the side of the triangle = 20 / 2 = 10
Now using pythagoras theorem we get,
\(20^2 \)= \(x^2\) + \(10^2\)
\(x^2\)= \(20^2 \)-\(10^2\)= 400 - 100= 300
\(x^2\)= 300
X = 10\(\sqrt{3}\)

Correct Answer: C

Approach Solution 3:

the altitude of an equilateral triangle creates two equal right angle triangles and divides the base into two portions of equal size
considering one of the right angle triangles,
- the side will form the hypotenuse => hypotenuse = 20
- the altitude will be the height of the triangle => altitude = x
- and the base will be half of the length of the original base of the equilateral triangle => base = 20/2 = 10

=> by Pythagoras theorem
altitude = √(\(20^2 \)-\(10^2\)) = 10√3

Correct Answer: C

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