Problem 54 - Algebra and Trigonometry

Question

Use slopes to show that \(A(-3,-1), B(3,3),\) and \(C(-9,8)\) are vertices of a right triangle.

Step-by-Step Solution

Answer

Yes, the slopes confirm a right triangle with a right angle at A.

Step 1: Calculate the Slope of AB

To find the slope between points A and B, use the formula: \( m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the coordinates for points A(-3, -1) and B(3, 3): \[ m_{AB} = \frac{3 - (-1)}{3 - (-3)} = \frac{4}{6} = \frac{2}{3} \]

Step 2: Calculate the Slope of BC

To find the slope between points B and C, use the same formula. Substitute the coordinates for points B(3, 3) and C(-9, 8):\[ m_{BC} = \frac{8 - 3}{-9 - 3} = \frac{5}{-12} = -\frac{5}{12} \]

Step 3: Calculate the Slope of AC

To find the slope between points A and C, use the formula. Substitute the coordinates for points A(-3, -1) and C(-9, 8):\[ m_{AC} = \frac{8 - (-1)}{-9 - (-3)} = \frac{9}{-6} = -\frac{3}{2} \]

Step 4: Check if Slopes are Perpendicular

To determine if two lines are perpendicular, their slopes must be negative reciprocals of each other. Check the product of slopes \(m_{AB}\) and \(m_{AC}\):\[ m_{AB} \times m_{AC} = \frac{2}{3} \times -\frac{3}{2} = -1 \]Since the product is -1, lines AB and AC are perpendicular at point A.

Key Concepts

Slope Calculation: The Basics

When working with right triangles in coordinate geometry, calculating the slope of each side is an essential step. The slope provides information about how steep a line is and its direction. To calculate the slope between two points, we use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the two points on a line.
It's important to maintain the order of subtraction, i.e., if you start with point 1 for the numerators, you need to do the same for denominators.
Calculating slopes helps us understand the relationships between the sides of the triangle and is a crucial step in proving that a triangle is a right triangle.

Understanding Perpendicular Lines

Perpendicular lines are lines that intersect to form four right angles. In terms of slopes, if two lines are perpendicular, the product of their slopes will be \(-1\). This is because the slopes of perpendicular lines are negative reciprocals of each other.
For two slopes \(m_1\) and \(m_2\), they are perpendicular if:

• \(m_1 \times m_2 = -1\)

In the world of coordinate geometry, establishing that two sides of a triangle are perpendicular is a key step in proving that the triangle is a right triangle.
Knowing how to find perpendicular lines using slopes not only helps with triangles but is a fundamental skill in geometry.

Coordinate Geometry: Connecting Points and Lines

Coordinate Geometry is a branch of geometry where we express geometric connotations in numerical expressions, using a coordinate plane. This plane is split into four quadrants and allows us to efficiently locate and connect points using their coordinates.
When working to prove a right triangle using coordinate geometry, focusing on calculating slopes of the sides helps in understanding the spatial relationships between points. The ability to determine perpendicularity from slopes is a powerful way to use coordinate points in geometrical proofs.

• It enhances visualization of geometric figures.
• Enables precise calculations and verification.

Utilizing coordinate geometry, you bridge the concepts of algebra and geometry, enabling a comprehensive method to solve complex geometric problems.