each side is therefore equal to 5.
if you draw an altitude from one of the vertices to the opposite side and perpendicular to that opposite side, you divide the triangle into 2 equal right triangles whose hypotenusse is equal to 5 and whose base is equal to 2.5 and whose altitude is equal to x because we don't know what it is.
by the pythagorean formula, we get:
a^2 + b^2 = c^2
in this triangle, c = 5, b = 2.5 and a = x
the formula becomes:
x^2 + 2.5^2 = 5^2
solve for x^2 in this equation to get:
x^2 = 5^2 - 2.5^2
this gets you x^2 = 18.75
take the square root of both sides of this equation to get:
x = sqrt(18.75) = 4.3301270129
there is another way to find this.
all the angles of the equilateral triangle are equal to 60 degrees.
when you drop the perpendicular, you get 2 right triangles that have a 90 degree angle and a 60 degree angle and a 30 degree angle.
this is what is commonly known as a 30-60-90 degree triangle.
the sine of 60 degrees is equal to opposite side divided by hypotenuse.
the hypotenuse is equal to 5.
the opposite side is the altitude which is equal to x.
the sine of 60 degrees is equal to sqrt(3)/2
you get the formula:
sin(60) = x/5
multiply both sides of this equation by 5 to get:
5*sin(60) = x
replace sin(60) with sqrt(3)/2 to get:
5*sqrt(3)/2 = x
simplify to get:
2.5*sqrt(3) = x
if you use your calculator to multiply 2.5 * sqrt(3), you will get:
x = 2.5*sqrt(3) = 4.330127019
that's the same as we got above, so both methods get you the same answer.
this is good.
your answer is:
x = 2.5*sqrt(3) which can also be shown as:
x = 4.330127019
here's a picture of your equilateral triangle.
