Problem 6 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If a point \(A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right), C\left(x_{3}, y_{3}\right)\) are the vertices of an equilateral triangle whose each side is equal to \(k\), then show that $$ \left|\begin{array}{lll} x_{1} & y_{1} & 2 \\ x_{2} & y_{2} & 2 \\ x_{3} & y_{3} & 2 \end{array}\right|^{2}=3 k^{4} $$

Step-by-Step Solution

Answer

The verification follows from triangular area properties. When squared, it matches \(3k^4\).

Step 1: Understand Problem Statement

We are given the vertices of an equilateral triangle \( A(x_1, y_1), B(x_2, y_2), C(x_3, y_3) \), with each side of the triangle equal to \( k \). We need to show that the determinant of the matrix formed by these coordinates turns out to be equal to \( 3k^4 \) when squared.

Step 2: Set up the Determinant

The determinant of the matrix is \( \left| \begin{array}{lll} x_1 & y_1 & 2 \ x_2 & y_2 & 2 \ x_3 & y_3 & 2 \end{array} \right| \). Calculating this determinant involves applying the rule of Sarrus or using the general formula for the determinant of a 3x3 matrix.

Step 3: Calculate the Determinant

Let's calculate the determinant: \[\text{Determinant} = \left( x_1(y_2 \cdot 2 - y_3 \cdot 2) - y_1(x_2 \cdot 2 - x_3 \cdot 2) + 2(x_2y_3 - x_3y_2) \right)\]This simplifies to:\[\text{Determinant} = 2(x_1y_2 + x_2y_3 + x_3y_1 - x_1y_3 - x_2y_1 - x_3y_2)\]

Step 4: Geometric Interpretation

The expression \( x_1y_2 + x_2y_3 + x_3y_1 - x_1y_3 - x_2y_1 - x_3y_2 \) represents twice the area of triangle \( ABC \) when divided by 2. Therefore, it simplifies to the determinant of the vertices forming triangle \( ABC \) divided by 2. This result squared must equate to \( 3k^4 \), corresponding to \[ 4 \times \text{Area of } \triangle^2 \].

Step 5: Evaluate the Expression

By the properties of an equilateral triangle, if the area is \( \frac{\sqrt{3}}{4} k^2 \), then squaring the determinant-based area expression and equating to \( 3k^4 \) verifies correctness: \[(\text{Area = } \frac{\sqrt{3}}{4} k^2 )^2 = 3 \left( \frac{k^2}{4} \right)^2 \]Using \( 4 \text{ times this correctly equates to } 3k^4 \).

Step 6: Finalize the Conclusion

Thus, when we calculate \( (\text{Area of } \triangle \times 4)^2 \) it equates correctly, aligning with the provided expression's exponentiated value of \( 3k^4 \). This completes the proof showing the condition based on the given geometric configuration.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number calculated from the elements of a square matrix. It helps understand many things about the matrix, including whether the matrix is invertible or not. For a 3x3 matrix like the one in our problem, the determinant can be found by a systematic approach. You can expand the formula for the determinant as follows:

• Multiply and subtract according to the diagonal rule of Sarrus for this 3x3 case.
• For the matrix \[\begin{pmatrix}x_1 & y_1 & 2 \x_2 & y_2 & 2 \x_3 & y_3 & 2 \end{pmatrix}\] this involves a systematic pattern of multiplying and combining different sets of elements from the matrix.

Applying the calculation for this specific matrix setup, you would end up with the expression in the simplified solution: \[2(x_1y_2 + x_2y_3 + x_3y_1 - x_1y_3 - x_2y_1 - x_3y_2)\]. Make sure to follow each step carefully to get to this result. Each element of the matrix contributes to this important process which underlies many geometric interpretations.

Area of a Triangle

The area of a triangle, in this context, is directly connected to the formula derived from the determinant. When vertex coordinates of a triangle are given as \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \), the area \( A \) of triangle ABC can be calculated using: \[A = \frac{1}{2} |x_1y_2 + x_2y_3 + x_3y_1 - x_1y_3 - x_2y_1 - x_3y_2|\].
This represents half of the absolute value of the determinant-like formula, providing a geometric interpretation of the matrix determinant.

• An important aspect of this calculation is understanding how the arrangement of points affects the sign and magnitude of the area computed.
• In our specific problem, it is equated and scaled to fit the properties of equilateral triangles known for their uniform angles and sides.

With uniform side length \( k \), adjustments ensure every computed area and determinant accommodates the specific geometric form of an equilateral triangle in the context.

Properties of Triangles

Triangles have several properties and classifications. An equilateral triangle is one particular type, known for having all three sides and internal angles equal. It is classified under special triangles due to its symmetry and simplicity. Here are some key properties:

• All internal angles are 60 degrees, making calculations more straightforward.
• The altitude line divides the triangle into two right triangles, assisting with problem simplification.
• Area can also be calculated with the formula \( \text{Area} = \frac{\sqrt{3}}{4} k^2 \)

For the given problem, every property works together to ensure that when calculating the determinant squared, the expression indeed yields \( 3k^4 \). This illustrates how known triangle properties tie into matrix determinants when analyzing structured geometric problems. Understanding these properties helps interpret and solve geometry-based algebra problems efficiently.