An inscribed circle of an equilateral triangle has radius of length 3. The ratio of the areas of the inscribed circle and the circumscribed circle is:

Question

An inscribed circle of an equilateral triangle has radius of length 3. The ratio of the areas of the inscribed circle and the circumscribed circle is:

 

a) 1:2

b) 1:3

c) 2:5

d) 1:4

e) none of these

 

I am not sure how to solve these. If someone could explain how to get the answers that would be amazing.

Answer 1

First draw a picture.  Start with an equilateral triangle.  Then draw the inscribed circle -- which touches the triangle at the midpoints of the three sides.  Then draw the circumscribed circle, which touches the triangle at each of its vertices.

 

Now draw the line segment from one of the vertices to the point opposite it where the triangle touches the inscribed circle.  Call the vertex A.

 

Finally, draw the line segment from the center of the circle to one of the other points where the inscribed circle touches the triangle.  Call the point B.

 

Now we have to use some facts that can be proven but that one could make a reasonable guess about just by looking at the picture.

 

1.  The two circles have the same center.  Call the center C.

2.  Angle CBA is a right angle.

3.  Line segment AC bisects angle A.

 

All of these can be proven, but I'm going to skip that here.  In solving this problem, you could make a reasonable guess that they are true and see where it leads.  Since this is a multiple choice problem, it if leads to one of the choices, then the very strong likelihood is that this choice is the one.

 

So now, since the vertex angles of an equilateral triangle are all 60°, angle CAB must be 30°.   And since CBA is a right angle, ΔABC is a right triangle.  Therefore, using a little trigonometry, CB / AC = sin 30° = 1/2.

 

So CB / AC = 1/2, and CB = 1/2 AC.

 

But notice that CB is the radius of the inscribed circle (call that length r), and AC is the radius of the circumscribed circle (call that length R).  So r = R/2.

 

The area of the inscribed circle is πr² and the area of the circumscribed circle is πR².  So the ratio of the areas of the inscribed circle and the circumscribed circle is πr² / πR² = r² / R² = (R/2)² / R² =

(R² / 4) / R² = 1/4.  So with a couple of reasonable guesses about the structure of the circles and triangle, we see that the answer is d.

 

Notice, by the way that we never had to use the fact that the the radius of the inscribed circle is 3.  The same ratio holds no matter what the actual lengths are.