Q.If area of triangle is 35 sq units with vertices $(2, -6)$, $(5, 4)$ and $(k, 4)$. Then $k$ is (A) 12 (B) $-2$ (C) $-12, -2$ (D) $12, -2$

Concept: Area of a triangle using coordinates — the absolute value of half the determinant of the vertices.

Step 1: Use the formula

Area $= \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$.

Step 2: Substitute $(2, -6)$, $(5, 4)$, $(k, 4)$:

$\frac{1}{2} \left| 2(4 - 4) + 5(4 - (-6)) + k(-6 - 4) \right| = 35$.

Step 3: Simplify:

$2(0) + 5(10) + k(-10) = 50 - 10k$. …

The key idea here is that the area of a triangle, when you know the coordinates of its three vertices, is half the absolute value of a certain determinant. This determinant essentially measures the "signed" area — the sign tells you the orientation of the points (clockwise or anticlockwise), but the actual area is always positive. So when a problem gives you the area, you set the absolute value of that expression equal to the given area, and that often yields two possible values for the unknown coordinate.

Let’s work through it.

1. Recall the formula for area using coordinates. For points $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, the area is:

This comes from the determinant of a $3 \times 3$ matrix. The expression inside the absolute value is the signed area (twice the actual area). The absolute value ensures we get a positive number.

2. Plug in the given vertices. Let $(x_1, y_1) = (2, -6)$, $(x_2, y_2) = (5, 4)$, and $(x_3, y_3) = (k, 4)$. Substitute into the formula:

3. Simplify inside the absolute value step by step.
   • First term: $2(4 - 4) = 2 \times 0 = 0$
   • Second term: $5(4 + 6) = 5 \times 10 = 50$
   • Third term: $k(-6 - 4) = k \times (-10) = -10k$ So the expression becomes:

4. Set this equal to the given area, 35 square units.

Multiply both sides by 2:

5. Solve the absolute value equation. An absolute value equation $|A| = B$ (with $B > 0$) means $A = B$ or $A = -B$. So:

• For the first: $50 - 10k = 70 \implies -10k = 20 \implies k = -2$
• For the second: $50 - 10k = -70 \implies -10k = -120 \implies k = 12$

6. Interpret the result. …

Method: Finding an Unknown Coordinate Given a Fixed Triangle Area (MCQ form)

Same technique as any "solve for the unknown vertex coordinate given the area," applied here to pick the correct option.

Steps

Step 1: Substitute the vertices (including the unknown) into the area formula

Step 2: Simplify to a linear expression in the unknown, inside the modulus

Combine the known numbers and isolate the term containing the unknown.

Step 3: Set the modulus equal to twice the given area and split into two cases …

Common Mistakes

Mistake 1: Solving only one branch of the modulus equation and picking an incomplete matching option

Why it's wrong: an equation like $|50-10k|=70$ has two solutions ($k=-2$ and $k=12$) — reporting only one misses the correct option if it lists both values together. Correct approach: always solve both the "$+$" and "$-$" cases of the absolute value equation before checking the options.

Mistake 2: Mishandling the sign when two of the given $y$-values are equal, causing a term to vanish incorrectly …