Let one of the ex-radii be r1. For an equilateral triangle, all 3 ex radii will be equal.
Area = r1 * (s-a), where 's' is the semi perimeter and 'a' is the side of the equilateral triangle.
∴ ex-radius of the equilateral triangle, r1 = As−aAs−a = 3√a23a2

Therefore, the ratio of these radii is a23√a23 : a3√a3 : 3√a23a2
Or the ratio is 1 : 2 : 3

From the sine rule of triangles we know asinAasinA = bsinBbsinB = csinCcsinC = k

Therefore, a = k(Sin A), b = k(Sin B) and c = k(Sin C)

Hence, we can rewrite cosAacosAa = cosBbcosBb = cosCccosCc as cosAkSinAcosAkSinA = cosAkSinBcosAkSinB = cosAkSinCcosAkSinC

Or Cot A = Cot B = Cot C
Or A = B = C
Or the triangle is an equilateral triangle

The sum of the interior angles of a triangle is 180o.

The question states that the largest of the angles is 70o.

Objective: Find the least possible value of the smallest of the three angles.

The least value for the smallest angle will be when the other two angles are as large as possible.

The largest value that the other two angles can take is 70o each.

Therefore, the least value of the smallest of the angles = 180 - 70 - 70 = 40o

The area of triangle ABD is half the area of triangle ABC.
Both triangles have the same base AB.
So, the altitude BD of triangle ABD will be half the altitude BC of triangle ABC.
So, z = y.

AD is the hypotenuse of the triangle ABD.
So, AD = sqrt(y^2+x^2).... (1)

In triangle ABC, AC2 = w2 = x2 + (y + z)2
But y = z, so we can rewrite the expression as w2 = x2 + (y + y)2
Or w2 = x2 + 4y2
Or x2 = w2 - 4y2

Substitute the value of x2 in (1): (y^2+w^2−4y^2)   = sqrt(w^2−3y^2)

Definition 1: All sides AND all angles of a regular polygon are equal.
Definition 2: A convex polygon is one in which the measure of all interior angles is less than 180o.
Key formula 1: Sum of the interior angles of a convex polygon with n sides = (n - 2) * 180
Key formula 2: Sum of the exterior angles of a convex polygon = 360o.

Let each exterior angle be Ao. Then each interior angle will be 120 + Ao.
The sum of an exterior and its corresponding interior angle is 180o.
Therefore, A + (120 + A) = 180 => A = 30o.

From key formula 2, sum of the exterior angles of a convex polygon = 360o.
i.e., if the polygon has 'n' sides, then n ×× A = 360
Number of sides of the regular polygon, n = 360measure of one exterior angle360measure of one exterior angle = 3603036030 = 12 sides.

   The correct answer is Choice (3). The polygon has 12 sides.

Choice (1) The point at which the three angle bisectors meet is the incentre and it is the centre of incircle that is drawn to any triangle. Choice (1) is true and is not the answer.

Choice (2) The median of any triangle will bisect a side, but it need not necessarily bisect the side at right angles. All 3 medians of an equilateral triangle and the median to the side that is not equal in an isosceles triangle meet the side at right angles. So, stating that median of any triangle will bisect the side at right angles is incorrect.

Therefore, choice (2) is the answer.

Choice (3) The 3 altitudes of a triangle meet concurrently at a point and the point is known as orthocentre. Choice (3) is true and is not the answer.

Choice (4) The 3 perpendicular bisectors of a triangle meet concurrently at a point and the point is known as circumcentre. Choice (4) is true and is not the answer

Let the side of the original square be 'a' units. Therefore, the area of the original square = a2 units.

Concept 1: The diameter of the circle of maximum possible dimension that is cut from the square will be the side of the square. So, its diameter will be 'a' units.
Concept 2:The diagonal of the square of maximum possible dimension that can be cut from the circle will be the diameter of the circle. So, the diagonal of the final square will be 'a' units.

If the diagonal of the final square is 'a' units, then its area = a22a22 units.
Therefore, the area of the new square will be 50% of the area of the original square.

   The correct answer is Choice (2).

The triangle which has its base as the length of the rectangle and its height as the width of the rectangle is the largest triangle that can be fitted in the triangle.

Alternatively, the triangle which has its base as the width of the rectangle and its height as the length of the rectangle will be the largest triangle that can be fitted in the rectangle.

The area computed in both the instances will be same.

If the base of the triangle is 'l' and its height 'w', then its area is lw2lw2 square units.
Alternatively, if the base of the triangle is 'w' units and its height is 'l' units, then its area is wl2wl2 square units.

   The correct answer is Choice (4).

The ant moves 20 times upward (vertically) and only 19 times forward (horizontally).

Therefore, the ant covers a distance of 20 * 0.5 = 10 feet when it moves upwards.
And moves 19 * 1 = 19 feet forward.

The total distance covered by the ant = 29 ft.

   The correct answer is Choice (4).

If the hour hand moved through 12 hours, it subtends 3600 at the centre of the clock.
i.e., in 12 hours it makes 3600.

Therefore, in 3 hours it makes 312312 x 360 = 900

The length of the hour hand is the radius of the circle.
We know that length of the hour hand = 7 cm. i.e., the radius of the circle is 7 cm.

Therefore, area of sector = 90036009003600 x π r2 = 1414 x 227227 x 7 x 7 = 38.5 sq.cm

   The correct answer is Choice (2).

If the hour hand moved through 12 hours, it subtends 3600 at the centre of the clock.
i.e., in 12 hours it makes 3600.

Therefore, in 3 hours it makes 312312 x 360 = 900

The length of the hour hand is the radius of the circle.
We know that length of the hour hand = 7 cm. i.e., the radius of the circle is 7 cm.

Therefore, area of sector = 90036009003600 x π r2 = 1414 x 227227 x 7 x 7 = 38.5 sq.cm

   The correct answer is Choice (2).