Problem 3 - Precalculus

Question

(a) Draw the right triangle \(P Q R\) with vertices \(P(1,0)\) \(Q(5,0),\) and \(R(5,3)\) (b) Use the formula for the area of a triangle, \(A=\frac{1}{2} b h\) to find the area of triangle \(P Q R\) in part (a).

Step-by-Step Solution

Answer

The area of triangle PQR is 6 square units.

Step 1: Plot the Points

First, plot the given points on a coordinate plane. The vertices of the triangle are \( P(1,0) \), \( Q(5,0) \), and \( R(5,3) \). This forms a right triangle with \( QR \) as the vertical leg and \( PQ \) as the horizontal leg.

Step 2: Identify the Base and Height

From the plotted points, determine the lengths of the base and height. The base is the horizontal distance between \(P(1,0)\) and \(Q(5,0)\), which is \( b = 5 - 1 = 4 \) units. The height is the vertical distance from \(Q(5,0)\) to \(R(5,3)\), which is \( h = 3 - 0 = 3 \) units.

Step 3: Use the Area Formula

Plug the base and height into the area formula for a triangle, \( A = \frac{1}{2} b h \). This gives \[ A = \frac{1}{2} \times 4 \times 3. \]

Step 4: Calculate the Area

Perform the multiplication to find the area. Simplifying \[ A = \frac{1}{2} \times 12 = 6. \] So, the area of triangle \( PQR \) is 6 square units.

Key Concepts

Triangle Area Formula

To find the area of a right triangle, we can use the triangle area formula, which is super simple. The formula is given by \[ A = \frac{1}{2} \times b \times h \] where:

• A is the area of the triangle.
• b is the length of the base, typically the horizontal side.
• h is the height, typically the vertical side.

To remember this, think of a full rectangle with the same base and height. The triangle is exactly half of that rectangle, hence the factor of \( \frac{1}{2} \). For a right triangle, identifying the base and height is straightforward, as they are the two perpendicular sides.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a way to describe geometric figures using a coordinate plane. This plane is divided into four quadrants by the x-axis and y-axis, with the point (0,0) called the origin.
A right triangle can easily be identified on a coordinate plane because it has one 90-degree angle formed by the intersection of one horizontal and one vertical line.
When plotting points, each point is defined by an \((x, y)\) pair. This helps in determining distances, such as the base and height of a triangle when given by their coordinates. For instance, the distance between points \(P(1,0)\) and \(Q(5,0)\) is simply \(5-1 = 4\) because they lie on the same horizontal line.

Plotting Points

Plotting points on a coordinate plane involves placing dots at specific locations based on their \((x, y)\) coordinates. Here's how you do it:

• Begin at the origin \((0,0)\).
• Move horizontally along the x-axis to the x-coordinate.
• Then, move vertically to reach the y-coordinate.
• Place a dot at the location defined by \((x,y)\).

For the right triangle in our exercise, the points are \(P(1,0)\), \(Q(5,0)\), and \(R(5,3)\). These points define a triangle where \(PQ\) is the horizontal line (base) and \(QR\) the vertical line (height). After plotting, you can easily visualize the triangle, making it simpler to apply the area formula.