Podcast
Questions and Answers
What is the outcome of applying the distance formula between points A(1, 2) and B(4, 6)?
What is the outcome of applying the distance formula between points A(1, 2) and B(4, 6)?
- $7$
- $6$
- $3$
- $5$ (correct)
Which of the following correctly defines the section formula for internal division?
Which of the following correctly defines the section formula for internal division?
- $P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$ (correct)
- $P\left( \frac{mx_2 - nx_1}{m-n}, \frac{my_2 + ny_1}{m+n} \right)$
- $P\left( \frac{mx_1 + nx_2}{m+n}, \frac{my_1 + ny_2}{m+n} \right)$
- $P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 - ny_1}{m-n} \right)$
If point P divides the line segment AB internally in the ratio 2:3, what is the x-coordinate of P when A(2, 3) and B(8, 5)?
If point P divides the line segment AB internally in the ratio 2:3, what is the x-coordinate of P when A(2, 3) and B(8, 5)?
- $5.2$
- $4.4$
- $5.8$
- $6$ (correct)
Which statement correctly describes external division in the context of the section formula?
Which statement correctly describes external division in the context of the section formula?
What is the correct formula for finding point P if points A(2, 5) and B(4, 9) are divided externally in the ratio 1:3?
What is the correct formula for finding point P if points A(2, 5) and B(4, 9) are divided externally in the ratio 1:3?
In which field is the distance formula NOT commonly used?
In which field is the distance formula NOT commonly used?
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Study Notes
Coordinate Geometry
Distance Formula
- Definition: Calculates the distance between two points in a coordinate plane.
- Formula: For points ( A(x_1, y_1) ) and ( B(x_2, y_2) ): [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
- Applications: Used in various fields such as physics, engineering, and computer graphics to find lengths between points.
Section Formula
- Definition: Determines the coordinates of a point dividing a line segment internally in a given ratio.
- Formula: For points ( A(x_1, y_1) ) and ( B(x_2, y_2) ) divided by point ( P ) in the ratio ( m:n ): [ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) ]
- Types:
- Internal Division: Point ( P ) lies between points ( A ) and ( B ).
- External Division: Point ( P ) lies outside the segment, with the formula adjusted accordingly as: [ P\left( \frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n} \right) ]
- Applications: Useful for locating points in geometric proofs, determining centroids in polygons, and in computer graphics for rendering.
Distance Formula
- Defines the method to calculate the distance between two points in a coordinate plane.
- Utilizes the formula: [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
- Widely applied in physics to measure space, in engineering for design accuracy, and in computer graphics for calculating dimensions between graphical elements.
Section Formula
- Determines the coordinates of a point that divides a line segment in a specified ratio, either internally or externally.
- Uses the formula for internal division: [ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) ]
- For external division, the formula alters to: [ P\left( \frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n} \right) ]
- Internal division finds points located between two given points, while external division identifies points outside the segment.
- Essential in geometric proofs, calculating centroids in polygons, and used in computer graphics to enhance rendering precision.
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