Q.The area of a triangle with vertices $(-3, 0)$, $(3, 0)$ and $(0, k)$ is $9$ sq. units. The value of $k$ will be
(A) $9$
(B) $3$
(C) $-9$
(D) $6$

The key idea is the area of a triangle using coordinates:

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

Step 1: Label the vertices:

$(x_1, y_1) = (-3, 0)$, $(x_2, y_2) = (3, 0)$, $(x_3, y_3) = (0, k)$.

Step 2: Substitute into the formula: …

Concept First: Why the Coordinate Formula Works

When you have three points on a plane, the area of the triangle they form can be found without drawing anything. The idea comes from the shoelace formula — you take the coordinates, multiply in a criss-cross pattern, and half the absolute difference. Why? Because each term $x_1 y_2 - x_2 y_1$ represents the signed area of a parallelogram formed by two vectors from the origin. Adding these for all three sides and halving gives the triangle’s area. The absolute value ensures area is positive.

For this problem, two vertices lie on the x-axis: $(-3,0)$ and $(3,0)$. That means the base of the triangle is along the x-axis, from $x = -3$ to $x = 3$, so the base length is $6$ units. The third vertex $(0,k)$ is directly above or below the midpoint of the base. So the height is simply $|k|$. This geometric shortcut is faster, but we’ll also do the full coordinate method to be thorough.

Step-by-Step Solution

1. Write down the coordinates.

Let $A = (-3, 0)$, $B = (3, 0)$, $C = (0, k)$.

2. Recall the area formula for a triangle given coordinates.

This formula works for any three non-collinear points.

3. Substitute the values.

Take $x_1 = -3$, $y_1 = 0$; $x_2 = 3$, $y_2 = 0$; $x_3 = 0$, $y_3 = k$.

Compute inside the absolute value:

• First term: $x_1(y_2 - y_3) = (-3)(0 - k) = (-3)(-k) = 3k$
• Second term: $x_2(y_3 - y_1) = (3)(k - 0) = 3k$
• Third term: $x_3(y_1 - y_2) = (0)(0 - 0) = 0$

Sum: $3k + 3k + 0 = 6k$

So the area is $\frac{1}{2} |6k| = 3|k|$.

4. Set the area equal to the given value.

We are told the area is $9$ square units. So:

Divide both sides by $3$:

Thus $k = 3$ or $k = -3$. …

Method: Area of a Triangle from Coordinates (with an Unknown Vertex)

When a triangle's area is given and one coordinate is an unknown, the coordinate area formula turns the area condition into a solvable equation for that unknown.

Steps

Step 1: Label the vertices

Write the three given points as $(x_1,y_1),(x_2,y_2),(x_3,y_3)$, keeping any unknown coordinate as a variable rather than substituting early.

Step 2: Substitute into the area formula

Simplify the expression inside the bars into a single expression in the unknown.

Step 3: Solve the resulting absolute-value equation

Set the simplified expression equal to $\pm$ the given area (an equation $|E| = k$ always splits into $E=k$ and $E=-k$), and solve both cases for the unknown. …

Common Mistakes

Mistake 1: Dropping the absolute value and solving $3k=9$ only

Why it's wrong: The area formula requires $|\cdot|$ because area can never be negative; solving without it gives only $k=3$ and silently discards the equally valid geometric solution $k=-3$ (which is also a listed option, (C)). Correct approach: always solve $|E|=\text{area}$ as two cases, $E=+\text{area}$ and $E=-\text{area}$, then check both against the answer choices.

Mistake 2: Mixing up which coordinate is $x$ and which is $y$ during substitution …