Problem 82 - Contemporary Precalculus

Question

Find the three points that divide the line segment from (-4,7) to (10,-9) into four parts of equal length.

Step-by-Step Solution

Answer

Question: Find three points that divide the line segment from (-4, 7) to (10, -9) into four parts of equal length. Answer: The three points are (-1/2, 3), (3, -1), and (6.5, -5).

Step 1: Recall the section formula

The section formula gives us the coordinates of a point dividing a line segment joining two points A(x1, y1) and B(x2, y2) in the ratio m : n. The formula is: ( (m*x2 + n*x1)/(m + n), (m*y2 + n*y1)/(m + n) )

Step 2: Apply the section formula for m = 1 and n = 3

Using the section formula, let's find the coordinates of the first point dividing the line segment in the ratio 1:3. A(-4, 7) and B(10,-9), m = 1 and n = 3: ( (1*10 + 3*(-4))/(1 + 3), (1*(-9) + 3*7)/(1 + 3) ) => ( (10 - 12)/4, (-9 + 21)/4 ) => (-1/2, 3) So, the first point is (-1/2, 3).

Step 3: Apply the section formula for m = 2 and n = 2

Now, let's find the coordinates of the second point dividing the line segment in the ratio 2:2. A(-4, 7) and B(10, -9), m = 2 and n = 2: ( (2*10 + 2*(-4))/(2 + 2), (2*(-9) + 2*7)/(2 + 2) ) => ( (20 - 8)/4, (-18 + 14)/4 ) => (12/4, -4/4) => (3, -1) So, the second point is (3, -1).

Step 4: Apply the section formula for m = 3 and n = 1

Finally, let's find the coordinates of the third point dividing the line segment in the ratio 3:1. A(-4, 7) and B(10, -9), m = 3 and n = 1: ( (3*10 + 1*(-4))/(3 + 1), (3*(-9) + 1*7)/(3 + 1) ) => ( (30 - 4)/4, (-27 + 7)/4 ) => (26/4, -20/4) => (6.5, -5) So, the third point is (6.5, -5).

Step 5: Check and display the points

We've found three points using the section formula: (-1/2, 3), (3, -1), and (6.5, -5). These are the points that divide the line segment from (-4, 7) to (10, -9) into four parts of equal length.

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of geometry where we use algebraic techniques to study geometric problems. It helps us to connect algebra and geometry using a system of coordinates. This system allows us to define geometric figures in a plane by using numerical values, typically Cartesian coordinates (x, y).

Here are some key aspects of coordinate geometry:

• Points are defined by pairs of numbers (x, y) that represent their position on a plane.
• Lines can be described by equations, making it easy to calculate properties like slope, length, and angles.
• Through analysis and equations, coordinate geometry aids in solving problems involving distance and midpoint calculations, lines, curves, and more complex shapes.

Coordinates are particularly useful for dividing or finding specific points on a line segment. By mastering coordinate geometry, students can solve a wide range of real-world problems effectively.

Ratio Division

Ratio division plays a crucial role in coordinate geometry to divide a line segment into specific pieces according to a given ratio. The section formula is a fundamental aspect, used specifically to find the coordinates of a point that divides a line segment between two endpoints into a particular ratio like m:n.

Key insights into ratio division include:

• Given two endpoints A(x₁, y₁) and B(x₂, y₂) of a line segment, the section formula \[\left( \frac{mx₂ + nx₁}{m+n}, \frac{my₂ + ny₁}{m+n} \right)\]provides the coordinates of the point dividing the line.
• The values of m and n indicate how far along the line segment the point will be placed. For instance, m:n = 1:1 places the point halfway, while m:n = 2:1 places it closer to the first endpoint.
• This technique is widely used in problems requiring division of segments like piecing, land surveying, etc.

Mastery of ratio division techniques expands one's ability to manage tasks involving line splitting in various disciplines.

Line Segment Division

Line segment division refers to the process of splitting a segment within a line into equal or specific parts using mathematical calculations. This concept is regularly utilized in coordinate geometry when points that divide the segment into equal segments must be calculated.

Breakdowns of key points include:

• Dividing a line segment can be done using equal segments or specific ratios depending on the requirements of the problem.
• The original exercise uses the section formula to find three points dividing the segment between (-4, 7) and (10, -9) into equal parts, exemplifying practical division.
• The formulas used accommodate both simple and complex division demands, supporting a variety of academic and practical tasks.

Whether for academics or real-world applications, understanding line segment division enhances precision and efficiency in spatial reasoning and geometric problem-solving.