Problem 32 - A Complete Resource Book in Mathematics for JEE Main

Question

\(A B\) is a diameter of a circle and \(C\) is any point on the circumference of the circle, then (A) area of \(\triangle A B C\) is maximum when it is an isosceles (B) area of \(\Delta A B C\) is minimum when it is an isosceles(C) the perimeter of \(\Delta A B C\) is minimum when it is isosceles (D) the perimeter of \(\Delta A B C\) is maximum when it is isosceles

Step-by-Step Solution

Answer

(A) The area is maximum when \( \triangle ABC \) is isosceles.

Step 1: Understanding the Geometry

Given a circle where \( AB \) is the diameter and \( C \) is any point on the circle, when \( A \), \( B \,\), and \( C \) are connected, triangle \( \triangle ABC \) is a right triangle (by the property of a semicircle). This is because the angle at \( C \) is a right angle.

Step 2: Evaluating Conditions for Area

The area of a triangle is given by \( \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \). For \( \triangle ABC \), \( AB \) is the base, and the height is maximized when \( C \) forms the perpendicular bisector of \( AB \). This happens when the triangle is a right-angled isosceles triangle, maximizing the area.

Step 3: Analyzing Triangle Perimeter

Perimeter of \( \triangle ABC = AB + AC + BC \). For triangle \( \triangle ABC \) with given \( AB \), an isosceles configuration does not necessarily minimize or maximize the perimeter. Instead, moving point \( C \) to be very close to either \( A \) or \( B \) would minimize the perimeter, since the remaining two sides would be very small, rather than when it is isosceles.

Step 4: Conclusion

Based on the analysis, the area of \( \triangle ABC \) is maximized when it is an isosceles triangle, and no definitive conclusion about the perimeter related to isosceles configuration. Therefore, the correct statement is A.

Key Concepts

Circle Geometry

Understanding circles is fundamental when solving problems related to geometry. A circle is a set of all points in a plane that are equidistant from a fixed point, called the center. The distance from the center to any point on the circle is the radius. The diameter is twice the length of the radius and passes through the center, effectively splitting the circle in half. This is important in circle geometry because any triangle formed by a diameter and another point on the circle will always be a right triangle. This special property is what makes many calculations easier in geometry problems involving circles.

Area of Triangle

Calculating the area of a triangle involves understanding its base and height. The formula for the area is:

• \[\text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height}\]

For a triangle with a fixed base, like in the case of triangle \( \triangle ABC \) where \( AB \) is the diameter, the area is determined by the height, which is the perpendicular distance from the point to the base. To maximize the area, you want the height to be as large as possible. Therefore, when point \( C \) is at a right angle, this forms an isosceles right triangle, maximizing the height, and subsequently the area.

Perimeter of Triangle

The perimeter of a triangle is simply the sum of the lengths of its three sides. For triangle \( \triangle ABC \), you calculate it as:

• \[\text{Perimeter} = AB + AC + BC\]

When considering how positioning point \( C \) affects the perimeter, note that in an isosceles triangle, two sides are equal. However, in our specific situation of \( \triangle ABC \), bringing \( C \) closer to \( A \) or \( B \) minimizes the sum of the lengths \( AC \) and \( BC \). This strategic movement makes the other sides smaller, thereby reducing the perimeter. Hence, an isosceles configuration does not necessarily relate to minimizing the perimeter in this case.

Right Triangle Properties

Right triangles have a special place in geometry because of their unique properties. Every right triangle has one 90-degree angle. When one leg is also the height (as in triangle \( \triangle ABC \) when point \( C \) is on the circle), it simplifies calculations for area using the designated height and base. The Pythagorean theorem is a key tool when dealing with right triangles. It states that:

• \[a^2 + b^2 = c^2\]

where \( c \) is the hypotenuse, the longest side opposite the right angle, and \( a \) and \( b \) are the other two sides. Using this theorem, you can determine unknown side lengths and make accurate calculations for both the area and perimeter, enhancing your problem-solving capabilities with right triangle properties.