Problem 67 - A Complete Resource Book in Mathematics for JEE Main

Question

The incentre of the triangle with vertices \((1, \sqrt{3}),(0,\), 0) and \((2,0)\) is: (A) \(\left(1, \frac{\sqrt{3}}{2}\right)\) (B) \(\left(\frac{2}{3}, \frac{1}{\sqrt{3}}\right)\) (C) \(\left(\frac{2}{3}, \frac{\sqrt{3}}{2}\right)\) (D) \(\left(1, \frac{1}{\sqrt{3}}\right)\)

Step-by-Step Solution

Answer

The incentre is not listed among the given options.

Step 1: Calculate the sides of the triangle

Label the vertices of the triangle as \( A(1, \sqrt{3}) \), \( B(0, 0) \), and \( C(2, 0) \). Calculate the lengths of sides \( BC \), \( CA \), and \( AB \) using the distance formula: \[ BC = \sqrt{(2-0)^2 + (0-0)^2} = 2 \] \[ CA = \sqrt{(2-1)^2 + (0-\sqrt{3})^2} = \sqrt{4} = 2 \] \[ AB = \sqrt{(1-0)^2 + (\sqrt{3}-0)^2} = 2 \] So, \( BC = 2 \), \( CA = 2 \), and \( AB = 2 \). The triangle is equilateral.

Step 2: Recognize triangle properties

Since all sides are equal \((2)\), the triangle is equilateral. In an equilateral triangle, the incentre is the same as the centroid, as well as the centre of the circumcircle.

Step 3: Calculate centroid coordinates

The formula for the centroid \((G)\) of a triangle with vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is: \[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \] Substitute \( x_1 = 1, x_2 = 0, x_3 = 2, y_1 = \sqrt{3}, y_2 = 0, y_3 = 0 \): \[ G = \left( \frac{1 + 0 + 2}{3}, \frac{\sqrt{3} + 0 + 0}{3} \right) = \left( 1, \frac{\sqrt{3}}{3} \right) \]

Step 4: Compare with given options

The calculation shows the incentre is at \( \left( 1, \frac{\sqrt{3}}{3} \right) \). Compare this with the given options, none of which match exactly. Recheck each step for any calculation errors.

Key Concepts

Equilateral Triangle

An equilateral triangle is a special kind of triangle where all three sides are of equal length. This also means that all internal angles are equal, each being 60 degrees. Due to its symmetrical nature, an equilateral triangle has unique properties:
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• All sides are congruent.
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• All angles are equal, specifically 60 degrees.
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• The medians, angle bisectors, and altitudes coincide, meaning they all meet at the same point, called the centroid or incentre.
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\In the exercise, we see a triangle with vertices at \( (1, \sqrt{3}), (0,0) \) and \( (2,0) \). By calculating the side lengths and finding them equal, we confirm that this triangle is equilateral.
\Understanding an equilateral triangle helps us determine other important points within the triangle like the incentre, centroid, and circumcentre which, in this case, all lie at the same point.

Distance Formula

The distance formula is crucial for calculating the length of a side in a triangle, especially when the vertices are given as coordinates on a plane. The formula is derived from the Pythagorean theorem, and it is used to find the distance between any two points \( (x_1, y_1) \) and \( (x_2, y_2) \) as:\[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]In our problem, the distance formula was used to confirm the equal side lengths of the triangle based on its vertices. This process involves simple arithmetic and substitution. It's important to handle square root calculations carefully to ensure accuracy.\Understanding the distance formula also allows us to grasp how all sides of an equilateral triangle are computed, reinforcing their equality and the triangle's inherent symmetrical properties.

Centroid of Triangle

The centroid of a triangle is a critical concept, particularly in equilateral triangles where it shares its location with the incentre and circumcentre. The centroid is the point where all three medians intersect. It acts as the "center of mass" of the triangle. The centroid \( G \) can be found using the formula:\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \]This formula takes the average of the coordinates of the vertices. Therefore, it's particularly straightforward for equilateral triangles, where symmetry simplifies calculation. \Applying this formula in the problem provides us a means to calculate the position of the incentre since they coincide in equilateral triangles.By understanding the role and calculation of the centroid, we gain insights into the geometric balance and central tendencies of triangles, especially useful in symmetric forms like equilateral triangles.