Question

For \(\mathrm{A}(7,-3,1)\) and \(\mathrm{B}(4,9,8)\), the point that divides \(\underline{\mathrm{AB}}\) from \(\mathrm{B}\) in the ratio \(2: 5\) is (A) \([(34 / 7),(39 / 7 \overline{),(42} / 7)]\) (B) \([(34 / 7),(39 / 7),\\{(-42) / 7\\}]\) (C) \([\\{(-34) / 7\\},(39 / 7),\\{(-42) / 7\\}]\) (D) \([\\{(-34) / 7\\},\\{(-39) / 7\\},\\{(-42) / 7\\}]\)

Short answer

The point that divides the line segment AB from point B in the ratio 2:5 is \((\frac{34}{7}, \frac{39}{7}, \frac{42}{7})\).

Step-by-step solution

Identify the given information

We have the coordinates of points A (7,-3,1) and B (4,9,8), and we need to divide the line segment AB from point B in the ratio 2:5.

Use the formula for dividing a line segment in a given ratio

The formula to find the point C that divides the line segment AB from point B in the ratio m:n is: \[C = (\frac{m * A_x + n * B_x}{m+n},\frac{m * A_y + n * B_y}{m+n},\frac{m * A_z + n * B_z}{m+n})\]

Substitute the given values

In our problem, the ratio is 2:5, so m = 2 and n = 5. Substituting the coordinates of A and B, and the values of m and n, into the formula, we get: \[C = (\frac{2 * 7+ 5 * 4}{2+5}, \frac{2 * (-3) + 5 * 9}{2+5}, \frac{2 * 1 + 5 * 8}{2+5})\]

Perform the calculations and find the coordinates of C

Perform the calculations inside the parentheses: \[C = (\frac{14 + 20}{7}, \frac{-6 + 45}{7}, \frac{2 + 40}{7})\] \[C = (\frac{34}{7}, \frac{39}{7}, \frac{42}{7})\]

Match the result with the given options

The coordinates of point C are \((\frac{34}{7}, \frac{39}{7}, \frac{42}{7})\), which matches with option (A). So, the point that divides the line segment AB from point B in the ratio 2:5 is \((\frac{34}{7}, \frac{39}{7}, \frac{42}{7})\).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, underpins many principles within mathematics, specifically geometry. It involves the study of geometry using a coordinate system and the principles of algebra. This fusion of algebra and geometry allows one to solve geometric problems by using equations and coordinates. The coordinates are numbers which represent points on a plane in relation to two axes, usually denoted as x (horizontal axis) and y (vertical axis). In three-dimensional space, a z-axis is also included.

In our exercise concerning the division of a line segment in a given ratio, coordinate geometry enables us to express the location of points A and B numerically and find the coordinates of a third point C, which divides the line segment AB in a specific ratio. This approach transforms a spatial problem into a numerical one, making it more manageable to solve using algebraic methods.

Section Formula

The section formula is a key tool in coordinate geometry for finding the coordinates of a point that divides a segment into a given ratio. It combines the concept of weighted averages with the coordinates of the endpoints of a segment. The formula is particularly useful for problems where the partitioning of the line segment is not into two equal halves (the midpoint), but any other ratio.

One common form of the section formula is the internal division formula, which is \[C = \left(\frac{{m \cdot A_x + n \cdot B_x}}{{m+n}}, \frac{{m \cdot A_y + n \cdot B_y}}{{m+n}}, \frac{{m \cdot A_z + n \cdot B_z}}{{m+n}}\right)\] where C is the point dividing the line connecting A and B in the ratio of m:n. Here, m represents the distance from point C to B, while n represents the distance from point C to A. Through this formula, you can precisely locate a point within or outside a line segment in 2D or 3D space.

Ratios and Proportions

Ratios and proportions are fundamental concepts in mathematics that express how two quantities relate to each other. A ratio is a comparison between two numbers showing the number of times one value contains or is contained within the other. For example, a ratio of 2:5 (read as '2 to 5') indicates that for every 2 units of the first quantity, there are 5 units of the second quantity. A proportion, on the other hand, states that two ratios are equal.

These concepts are crucial when it comes to dividing a line segment in a given ratio; they enable us to determine how much of the line segment each point 'controls.' This relationship is used in coordinate geometry to distribute the lengths in line segments and apply to the section formula accordingly, ensuring accuracy in finding a dividing point.

Geometry Problems

Geometry problems in mathematics often involve understanding shapes, sizes, relative positions of figures, and the properties of space. When geometry problems are approached through the lens of coordinate geometry, they require an understanding of the coordinate plane, plotting points, and using algebraic methods to solve for unknowns.

The exercise we looked at is a classic example of a geometric problem tailored for coordinate geometry techniques. It involves understanding the concept of a line segment, how to express it in terms of coordinates, and how to apply algebraic formulas like the section formula. Solving geometry problems like this not only reinforces the understanding of geometry concepts but also enhances algebraic skills and problem-solving techniques.