What Is The Length Of The Side Of An Equilateral Triangle If Its Altitude Is 18 Cm?

What is the length of the side of an equilateral triangle if its altitude is 18 cm?

 

Answer:

The length of the side of the equilateral triangle is 12√3  cm.

Please click the image below to view my illustration and the complete solution.

Step-by-step explanation:

The altitude of an equilateral triangle (60°-60°-60°) divides it into two congruent 30°-60°-90° special right triangle.

The new triangle formed has  a short (opposite the 30°), longer leg (opposite the 60°) and the hypotenuse (opposite the 90°). The hypotenuse is the length of the original equilateral triangle.

The theorem of 30°-60°-90° special right triangle states that the hypotenuse is twice the shorter leg and the longer leg is the product of shorter leg and √3. The longer leg is also half the product of the hypotenuse and √3.

The altitude of the original equilateral triangle is also the longer leg of the new special triangle formed, while the hypotenuse is the side of the original equilateral triangle.

Find the length of the hypotenuse or the side of the equilateral triangle:

Given:

Longer leg: 18 cm

Solution:

Longer leg = (1/2)(√3)(hypotenuse)

18 = (1/2)(√3)(hypotenuse)

Hypotenuse = 18 (2/√3)

Hypotenuse = 36/√3

Hypotenuse = (36√3)/(√3)²  ⇒  Rationalize to remove a radical denominator

Hypotenuse = (36√3)/3

Hypotenuse = 12√3  cm.

Hypotenuse = Side of the equilateral triangle

Therefore:

The length of the side of the equilateral triangle is 12√3  cm.