An equilateral triangle has sides of 20. What are the lengths of another equilateral triangle with half the area?

Question

Evelyn Arboleda · Accepted Answer

#10sqrt2#

Area of triangle# =1/2a b sinC#

for this equilateral triangle

#A_1=1/2xx20xx20xxsin60#

#A_1=100sqrt3#

now we have the relation

#"ratio of areas " =(" ratio of lengths ")^2#

let the second triangle have area

#A_2#

#A_2=1/2xx100sqrt3=50sqrt3#

so #A_2:A_1::(l_2:20)^2#

#:. (50sqrt3)/(100sqrt3)=l_2^2/400#

#=>l_2^2=1/2xx400#

#l_2=sqrt(200)#

#l_2=10sqrt2#

Allison Allen · Answer

undefined The area of an equilateral triangle can be calculated using the formula $ \frac{\sqrt{3}}{4} \times \text{side}^2 $.

Given that the original equilateral triangle has sides of length 20, its area would be $ \frac{\sqrt{3}}{4} \times 20^2 $.

To find another equilateral triangle with half the area, we can set up the equation:

$ \frac{\sqrt{3}}{4} \times \text{new side}^2 = \frac{1}{2} \times \left( \frac{\sqrt{3}}{4} \times 20^2 \right) $

Solve this equation to find the length of the side of the new equilateral triangle.

$ \text{new side}^2 = \frac{\frac{\sqrt{3}}{4} \times 20^2 \times 2}{\frac{\sqrt{3}}{4}} $

$ \text{new side}^2 = 20^2 \times 2 $

$ \text{new side}^2 = 400 \times 2 $

$ \text{new side}^2 = 800 $

$ \text{new side} = \sqrt{800} $

$ \text{new side} \approx 28.28 $

Thus, the lengths of the sides of another equilateral triangle with half the area would be approximately 28.28 units.