Textbook Question
For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the midpoint M of line segment PQ. See Examples 1 and 2. P(-5, -6), Q(7, -1)
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Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem 13
For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the midpoint M of line segment PQ. See Examples 1 and 2.
P(8, 2), Q(3, 5)
Verified step by step guidance1
Identify the coordinates of points P and Q. Here, P has coordinates (8, 2) and Q has coordinates (3, 5).
To find the distance between P and Q, use the distance formula: \(d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of P and Q respectively.
Substitute the coordinates into the distance formula: \(d(P, Q) = \sqrt{(3 - 8)^2 + (5 - 2)^2}\).
To find the midpoint M of the line segment PQ, use the midpoint formula: \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\).
Substitute the coordinates of P and Q into the midpoint formula: \(M = \left( \frac{8 + 3}{2}, \frac{2 + 5}{2} \right)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Distance Formula
The distance formula calculates the length between two points in a plane using their coordinates. It is derived from the Pythagorean theorem and given by d = √[(x2 - x1)² + (y2 - y1)²]. This formula helps find the straight-line distance between points P and Q.
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Quadratic Formula
Midpoint Formula
The midpoint formula finds the point exactly halfway between two given points in a coordinate plane. It is calculated by averaging the x-coordinates and y-coordinates separately: M = ((x1 + x2)/2, (y1 + y2)/2). This gives the coordinates of the midpoint M of segment PQ.
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Coordinate Geometry Basics
Coordinate geometry connects algebra and geometry by representing geometric figures using coordinates. Understanding how points, lines, and distances are expressed in the coordinate plane is essential for applying formulas like distance and midpoint effectively.
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