1 Answer

MrQWERTY · Answer 1 · 2023-03-28T05:03:55+0000

Answer:

A=(√(3))/(4)a^2

Explanation:

Since an equilateral has all of the sides equal, we can find the height of triangle using:
a^2+b^2=c^2 . I attached a diagram which should explain how I got the dimensions of the three sides. Using the information from the diagram we get the equation:

h^2+((a)/(2))^2=a^2

Subtract a^2 from both sides

h^2=a^2-((a)/(2))^2

Take the square root of both sides

$h = \sqrt{a^2-((a)/(2))^2}$

If you know the area of a triangle, it's:
(1)/(2)bh . In this case the base=a, and the height is what we defined above. Using this we get:

$A = (a)/(2)*\sqrt{a^2-((a)/(2))^2}$

We can distribute the exponent over the division to get:

$A = (a)/(2)*\sqrt{a^2-((a^2)/(4))$

Now we can rewrite a^2 as 4a^2/4

$A = (a)/(2)*\sqrt{(4a^2)/(4)-((a^2)/(4))$

Now add the two fractions:

$A = (a)/(2)*\sqrt{(3a^2)/(4)$

We can distribute the square root the division just like how we distributed the exponent 2, since the square root can be expressed as an exponent (1/2)

A = (a)/(2)*(√(3a^2))/(√(4))

There's a radical identity that states:
$\sqrt[n]{a} * \sqrt[n]{b} = \sqrt[n]{a*b}$ . We can use this to rewrite one radical as multiple radicals to simplify it: