Incircle / triangle question and circumcircle?

I had seen the question of your son when posted and also the answer given by gohpihan which was perfect. So I did not post any answer and just left giving TU to him.

Now, I find that his answer is selected as BA which is in order, but could not follow the remarks posted by him that the question is incorrect.

I shall think over what you have now asked and post my reply as soon as possible.

Edit:

Yes, I also notice that there is a flaw in the question and as a result using the formula gives an incorrect answer which is a hypothetical answer.

I also recall that when I used to teach coordinate geometry, this type of problem used to occur. To find the equation of a circle passing through the points of intersection of the two circles S1 = 0 and S2 = 0 and also through an additional point (x1, y1), we use the method writing that any circle passing through the points of intersection of the circles is given by S1 + kS2 = 0. Then substituting the coordinates (x1, y1), find k and plugging k in the equation should give the equation of the required circle. But the problem will arise when S1 = 0 and S2 = 0 are not at all intersecting. In that case, one will get the answer following the general prescribed method, but the answer will represent a hypothetical circle. The above case is also similar.

Now, I will think over the question raised by you and post if I can find a solution.

Interesting question in any case!!

Minimum area of a triangle in which a incircle with radius 8 cm can be drawn

= 192√3 sq. cm. worked out as under.

Of all the triangles drawn for the incircle with radius 8 cm, the equilateral triangle will have the least area.

Let a/2 = half the length of side of the equilateral triangle

=> (a/2) / r = cot30°

=> a/2 = 8√3 cm

=> minimum area of the triangle

= 3 * (a/2) * 8 sq. cm.

= 192√3 sq. cm.

≈ 332.6 sq. cm.

Edit:

In terms of r,

minimum area

= 3 * √3r * r

= 3√3 r^2

=> if the area is maintained at 200 cm^2 in the question,

r < √[200/(3√3)] cm = 6.20 cm.

Edit:

You mentioned that this is a question for pre engg preparation and wanted to know what the student should answer here. I believe that the person who designed the question may have failed to understand the flaw in the question. One can answer that no circle with area 200 cm^2 and inradius 8 cm is possible by proving the formula for minimum area of the circle in terms of its in-radius as above. Moreover, in the above question it can easily be proved that the question is incorrect by finding the area of the circle of radius 8 cm and showing it to be more than the given area. Detailed answer, proving the connection between the minimum area and radius, will be necessary only if the radius is given as 7 cm in which case the area of the circle will be less than the given area, but still it would be an impossibility.

Edit:

b)

I just noticed this question as different from your original querry.

This too can be answered similarly.

For minimum area of circumcircle, we have to find the minimum value of circumradius, R

Minimum R

= (p/3) / 2sin60, where p = perimeter ... [Using the formula a/sinA = 2R]

= p / (3√3).

=> minimum area of a circumscribing circle for a given perimeter, p

= π(Rmin)^2

= πp^2/(27).

1. Inradius r = area of triangle/semiperimeter = [ {{3^1/2}a^2}/4 ]/[3a/2] = (3^1/2)a/2 So, area = 3pi*a^2/4 2. Circumradius = Product of sides/(4*Area of triangle) = a/(3^1/2) So, area = pi*a^2/3