Problem 80 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a, b, c\) are the sides of a triangle \(A B C\) such that \(\left|\begin{array}{ccc}a^{2} & b^{2} & c^{2} \\ (a+1)^{2} & (b+1)^{2} & (c+1)^{2} \\ (a-1)^{2} & (b-1)^{2} & (c-1)^{2}\end{array}\right|=0\), then \(\Delta A B C\) is (A) a right angled triangle (B) an isosceles triangle (C) an equilateral triangle (D) None of these

Step-by-Step Solution

Answer

(B) The triangle is isosceles.

Step 1: Identify the problem

We need to determine the type of triangle ABC, given that the determinant of a 3 x 3 matrix involving its sides is zero.

Step 2: Analyze the determinant condition

The determinant of the matrix being zero implies linear dependence among the rows (vectors), which suggests that there might be some condition regarding the equality or proportionality of the elements involved.

Step 3: Understand matrix structure

The matrix is composed of three sequences: \(a^2, b^2, c^2\), \((a+1)^2, (b+1)^2, (c+1)^2\), and \((a-1)^2, (b-1)^2, (c-1)^2\). This suggests a relation to the sides of a triangle but also implies symmetries of transformation.

Step 4: Solve determinant setup

Given \\[\left|\begin{array}{ccc}a^{2} & b^{2} & c^{2} \ (a+1)^{2} & (b+1)^{2} & (c+1)^{2} \ (a-1)^{2} & (b-1)^{2} & (c-1)^{2}\end{array}\right| = 0\] \we interpret this as the rows being linearly dependent, implying some sides must be equal or follow a special rule.

Step 5: Consider specific triangle types

Since the determinant relates to sides in a symmetric form, consider cases where values are equal. We check permutations where at least two sides are equal: \(a = b\), \(b = c\), or \(c = a\).

Step 6: Conclude on triangle type

The matrix conditions lead us to conclude a symmetrical requirement among the lengths. Thus, the configuration satisfies an isosceles triangle condition, where at least two sides are equal.

Key Concepts

Triangle Properties

Triangles are fundamental shapes in geometry and have various properties that define their types and characteristics. In general, a triangle is a three-sided polygon with three angles. The sum of these angles is always 180 degrees. Understanding the properties of triangles helps in determining their classification based on side lengths and angles.

There are several types of triangles based on angles:

• Acute triangle: All angles are less than 90 degrees.
• Right triangle: One angle is exactly 90 degrees.
• Obtuse triangle: One angle is more than 90 degrees.

Triangles are also classified based on side lengths:

• Scalene triangle: All sides are of different lengths.
• Isosceles triangle: At least two sides are equal in length.
• Equilateral triangle: All three sides are equal in length and all angles are 60 degrees.

Recognizing these properties is vital in solving geometric problems and determining the specific type of triangle in question.

Isosceles Triangle

An isosceles triangle is one where at least two of its sides are equal in length. This equality not only affects the sides but also the angles opposite these sides. The angles opposite the equal sides are equal, leading to some unique properties of isosceles triangles.

Some important aspects of isosceles triangles include:

• Symmetry: An isosceles triangle has an axis of symmetry along the perpendicular bisector of the unequal side.
• Equality of angles: If two sides are equal, the angles opposite these sides are also equal.
• Height and area: The height from the apex (where the two equal sides meet) to the base (unequal side) bisects the base. This height is crucial for calculating the area of the triangle.

These properties make isosceles triangles a frequent subject of geometric proofs and calculations, as they provide simplicity and symmetry in problem-solving.

Matrix Conditions

The given exercise involves conditions on a matrix that lead to conclusions about the triangle's characteristic. A determinant of a matrix indicates whether the rows (or columns) of the matrix are linearly dependent or independent.

In this case, the matrix is:
\[\begin{array}{ccc} a^{2} & b^{2} & c^{2} \ (a+1)^{2} & (b+1)^{2} & (c+1)^{2} \ (a-1)^{2} & (b-1)^{2} & (c-1)^{2} \end{array}\]
The determinant being zero means that the rows of this matrix are linearly dependent. This linear dependence suggests a specific relationship among the sides of the triangle.

The relationships possible are generally indicative of equalities or proportionalities between the elements. When applied to the context of triangles, it often signals conditions like that of an isosceles triangle, where a pair of elements (sides) are equal. Hence, by evaluating these matrix conditions and their implications, we can conclude on the type of triangle based on its sides.