Question 1204108: Find the radius of the circle inscribed in an equilateral triangle whose perimeter is 10.8 units Found 2 solutions by ikleyn, math_tutor2020:Answer by ikleyn(52799) (Show Source):
You can put this solution on YOUR website! .
Find the radius of the circle inscribed in an equilateral triangle whose perimeter is 10.8 units.
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For any triangle, from consideration its area, it is easy to deduce the formula
for the radius "r" of the inscibed circle
= area. (1)
where P is its perimeter.
For the given equilateral triagle, P = 3a, area = , where "a" is the side length,
a = = 3.6 units.
Therefore, from (1)
= .
It implies r = = = = = 1.03923 (rounded).
ANSWER. The radius of the inscribed circle is r = = = 1.03923 (rounded).
You can put this solution on YOUR website!
This is what the diagram could look like
A,B,C = vertices of the equilateral triangle
D = center of the inscribed circle
E = midpoint of AB
Triangle ADE is a right triangle. More specifically, it is a 30-60-90 triangle.
This is because angle CAB = 60 is bisected to help form angle DAE = 30 degrees.
Also, angle ADE = 60 degrees.
The perimeter of the equilateral triangle is 10.8 units.
Each side must be (10.8)/3 = 3.6 units
AB = 3.6
BC = 3.6
AC = 3.6
Because E is the midpoint of AB, we then know
AE = AB/2 = (3.6)/2 = 1.8
Segment AE is the longer leg of the 30-60-90 triangle ADE (notice it's opposite the 60 degree angle).
For any 30-60-90 triangle we have this template
In this particular case it means
Isolating DE gets us
Which is the approximate radius of the inscribed circle.
Round this value however needed. Or you can stick to the exact value to avoid worrying about rounding.
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