Trig question i'm confused on?

Question

If x denotes the length of a side of an equilateral triangle express teh area of the trinagle as a function of x.

I need the steps on how to figure this out. Thanks!

Answer 1

Area = (1/2) Base x Height.

Base is x, because you can choose any side as base. All are equal.

The Height (h), half of Base (0.5x), and one Side(x) form a right angled triangle.

h/x = sin 60 = sqrt (3)/2

h = (sqrt(3)/2)x.

Area = (1/2)x (sqrt(3)/2)x
=sqrt(3)*x^2/4

Answer 2

Let the eqilateral triangle be ABC where AB is the base and a line drawn from C perpendicular to AB at point D is the height or altitude of triangle ABC.

The altitude of an equilateral triangle bisects the angle C and also bisects the base so that AD = BD.

Each angle of an equilateral triangle is =60 degrees. Therefore the right triangle ACD is a 30-60-90 degree triangle. In thes triangles the following relationships hold:
The hypotenuse is twice the shorter leg, and
the longer leg = shorter leg*sqrt(3).

In our triangle the longer leg is (x/2)sqrt(3) which is the altitude.
The area A is 1/2 the product of the base and altitude , so
A = (1/2) (x) (x/2) sqrt(3) = [(x^2*sqrt(3)]/4

Answer 3

Equilateral triangles always have 60 degrees to each side. So if you can imagine a equilateral triangle and set in on a side. Bisect the top angle and youll create two right triangles. the bottom side is also bisected so it is now .5x. From there you can use the pythagorean theorem to figure out the height in terms of x ( which is the square root of .75 times x. And now that you have the base and height you can find the area of the triangle ... .5* (.75)^(1/2) * x^2

Answer 4

all sides of the triangle are x. and the formula for area of a triangle is bh/2.
b=x=one of the sides.
the height can be solved by dropping a purpendicular line from one of the angles to the oposite side. the hyp will be x, one side will be 1/2x (because the perpendicular will bisect a side because 30(the bisected angle),60 (the original angle),90(the perpendicular meeting the oposite side) and the third side (the height) can be found to be [sqrt(3)/2]x by the pythagorean theorem. So, b=x and height=[sqrt(3)/2]x

area of an equilater triangle is [sqrt(3)/4]x^2

Answer 5

One thing you need to know about equilateral triangles is that each angle is aslo equivalent making each angle equal 60 degrees

The height of the triangle will therefore be xsin60
Area = bh/2
=(x)(xsin60)/2
=x^2sin60/2
sin60 = sqrt(3)/2
after subbing that into sin60:
Area = x^2sqrt(3)/4

Answer 6

Draw any altitude. You now have two 30-60-90 right triangles whose hypotenuses = x. The short leg is obviously x/2, and pythagoras tells you the altitude is √3/2*x. Now you have the base & altitude, and I assume you know the formula for the area of a triangle..........

Answer 7

one way could be to analyze a diagram of a unit circle and observe the places the place sinx=cosx are pi/4 and 5pi/4 (40 5 levels and 225 levels), then set those angles equivalent to 2x and resolve for x. it incredibly is how i might resolve it, yet you'll be able to additionally manage one area of the equation (like somebody above me did) to get the two in terms of the comparable trig function, then set the interior angles equivalent. bear in techniques, nevertheless, that as quickly as you are trying this you purely get between the angles, whilst the equation has 2 suggestions.

Answer 8

draw the triangle... the sides are all equal... the angles are all equal... the sides are all x... find the hieght the hieght is then the sin of 60 times x.... multiply the hieght by the base (x/2) and then divied that answer by two

Answer 9

the wording basically means
whats the area in terms of x?

area for equilateral triangle = (x^2root(3))/4
so thats ur answer

Answer 10

x^2 * sqrt(3) / 4