Problem 58 - Algebra and Trigonometry

Question

Find the area of the triangle that lies in the first quadrant (with its base on the \(x\) -axis) and that is bounded by the lines \(y=2 x-4\) and \(y=-4 x+20\).

Step-by-Step Solution

Answer

The area of the triangle is 6 square units.

Step 1: Find the Intersection Points

To find the intersection points of the two lines, set the equations equal to each other: \(2x - 4 = -4x + 20\). Solving for \(x\), add \(4x\) to both sides and add \(4\) to both sides to get: \(6x = 24\). Divide both sides by \(6\) to find \(x = 4\). Substitute \(x = 4\) into one of the equations, say \(y = 2x - 4\), to find \(y = 2(4) - 4 = 4\). Thus, the point of intersection is \((4, 4)\).

Step 2: Find the Intersection with the X-axis

The intersections with the \(x\)-axis occur when \(y = 0\). For \(y = 2x - 4\), set \(2x - 4 = 0\), which gives \(x = 2\). For \(y = -4x + 20\), set \(-4x + 20 = 0\), which gives \(x = 5\). Thus, the triangle's base on the \(x\)-axis extends from \((2, 0)\) to \((5, 0)\).

Step 3: Calculate the Base and Height of the Triangle

The base of the triangle is the distance between the points \((2, 0)\) and \((5, 0)\), which is \(5 - 2 = 3\). The height of the triangle is the \(y\)-coordinate of the point where the lines intersect in the first quadrant, which is \(4\).

Step 4: Calculate the Area of the Triangle

Use the formula for the area of a triangle, \( \frac{1}{2} \times \text{base} \times \text{height} \). Plug the values you found: \( \frac{1}{2} \times 3 \times 4 = 6\). Thus, the area of the triangle is \(6\) square units.

Key Concepts

Intersection Points

Finding the intersection points of lines is a crucial step in understanding where two lines might meet on a graph. In this exercise, two linear equations, \(y = 2x - 4\) and \(y = -4x + 20\), are provided. By setting these equations equal, we solve for the \(x\)-coordinate of the point where these lines intersect.
Here's how you calculate it:

• Set \(2x - 4 = -4x + 20\).
• Add \(4x\) to both sides to get \(6x - 4 = 20\).
• Add \(4\) to both sides to simplify to \(6x = 24\).
• Divide by \(6\) to find \(x = 4\).

Once \(x\) is determined, substitute \(x = 4\) into either original equation to find the \(y\)-coordinate. Using \(y = 2x - 4\), substitute \(x = 4\) to get \(y = 4\). Thus, the intersection point is \((4, 4)\). This point is essential as it represents the vertex opposite the triangle's base.

Base and Height of a Triangle

To calculate the area of a triangle, understanding the base and height is fundamental. In this scenario, the triangle's base lies along the \(x\)-axis between the intersection points of each line with the axis.
These points are where the lines intersect the \(x\)-axis; mathematically set \(y = 0\) to solve for these points:

• For \(y = 2x - 4\), setting \(y = 0\) gives \(2x = 4\), so \(x = 2\).
• For \(y = -4x + 20\), setting \(y = 0\) gives \(-4x = 20\), so \(x = 5\).

The triangle's base thus extends from \((2, 0)\) to \((5, 0)\) on the \(x\)-axis. To find this base length, simply subtract the \(x\)-values: \(5 - 2 = 3\).
The height is more direct, as it is the \(y\)-coordinate of the intersection point found earlier, which is \(4\). Knowing these two measurements will allow accurate calculation of the triangle's area.

Area Calculation

Once the base and height of the triangle are known, calculating the area becomes straightforward with the formula for the area of a triangle: \[\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\]From our earlier findings, the base of the triangle is \(3\) units and the height is \(4\) units. Plug these values into the formula:

• Area \(= \frac{1}{2} \times 3 \times 4\)
• This simplifies to \(\frac{1}{2} \times 12\)
• The result is an area of \(6\) square units.

This formula is a basic yet powerful tool in geometry. It highlights the importance of accurately determining the base and height, which are the keystones for calculating the area efficiently.