Finding the Value of 'a' for a Triangle's Area

The problem provides us with three vertices of a triangle and its area. Two vertices are given directly, and the third vertex is the point where two lines intersect. Our goal is to use the area formula for a triangle with given vertices to find the unknown value 'a'.

Identifying the Triangle Vertices

The first two vertices are given as (3, 4) and (5, 2). Let's call these:

• Vertex 1: \((x_1, y_1) = (3, 4)\)
• Vertex 2: \((x_2, y_2) = (5, 2)\)

The third vertex is the point of intersection of the lines \(x = a\) and \(y = 5\). A point on the line \(x = a\) has an x-coordinate equal to 'a'. A point on the line \(y = 5\) has a y-coordinate equal to 5. The point of intersection must satisfy both conditions. Therefore, the third vertex is:

• Vertex 3: \((x_3, y_3) = (a, 5)\)

The area of this triangle is given as 3 square units.

Area Calculation using Vertex Coordinates

The area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the formula:

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

We are given that the Area is 3 square units. Substituting the coordinates of our vertices into the formula:

\(3 = \frac{1}{2} |3(2 - 5) + 5(5 - 4) + a(4 - 2)|\)

Solving for 'a' using the Area Formula

Now, let's simplify the expression inside the absolute value:

Multiply both sides by 2 to get rid of the fraction: \(2 \times 3 = |3(2 - 5) + 5(5 - 4) + a(4 - 2)|\) \(6 = |3(-3) + 5(1) + a(2)|\)

Perform the multiplications: \(6 = |-9 + 5 + 2a|\)

Combine the constant terms: \(6 = |-4 + 2a|\)

We can rewrite this as: \(6 = |2a - 4|\)

The absolute value equation \(|X| = k\) implies \(X = k\) or \(X = -k\). So, we have two possible cases:

Case 1: \(2a - 4 = 6\)

Add 4 to both sides: \(2a = 6 + 4\) \(2a = 10\)

Divide by 2: \(a = \frac{10}{2}\) \(a = 5\)

Case 2: \(2a - 4 = -6\)

Add 4 to both sides: \(2a = -6 + 4\) \(2a = -2\)

Divide by 2: \(a = \frac{-2}{2}\) \(a = -1\)

We have found two possible values for 'a': 5 and -1. We need to check which of these values is present in the given options.

Comparing with Options

The given options for the value of 'a' are:

• 2
• 3
• 4
• 5

Comparing our calculated values (5 and -1) with the options, we see that 5 is one of the options, while -1 is not.

Therefore, the value of a is 5.

Revision Table: Key Concepts for Triangle Area

Additional Information: Alternate Area Calculation Methods

Besides the coordinate formula, there are other ways to calculate the area of a triangle:

• Base and Height: Area \( = \frac{1}{2} \times \text{base} \times \text{height}\). This requires finding the length of a base and the perpendicular distance from the opposite vertex to that base (the height).
• Heron's Formula: If the lengths of all three sides (sidelengths A, B, C) are known, Area \( = \sqrt{s(s-A)(s-B)(s-C)}\), where \(s\) is the semi-perimeter \(s = \frac{A+B+C}{2}\). This method requires calculating the distances between all three pairs of vertices first.
• Using Determinants: The area can also be found using a determinant: Area \( = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|\). This is equivalent to the formula used in the solution.