Textbook Question
CONCEPT PREVIEW Perform the operations mentally, and write the answers without doing intermediate steps. √25 + √64
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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 7
CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. The midpoint of the segment joining (0, 0) and (4, 4) is ________.
Verified step by step guidance1
Recall that the midpoint of a segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[\left( \frac{\,x_1 + x_2}{2}, \frac{\,y_1 + y_2}{2} \right)\]
Identify the coordinates of the two given points: \((0, 0)\) and \((4, 4)\). Here, \(x_1 = 0\), \(y_1 = 0\), \(x_2 = 4\), and \(y_2 = 4\).
Substitute these values into the midpoint formula:
\[\left( \frac{0 + 4}{2}, \frac{0 + 4}{2} \right)\]
Simplify the expressions inside the parentheses by performing the addition and division:
\[\left( \frac{4}{2}, \frac{4}{2} \right)\]
Express the simplified midpoint coordinates as a point:
\[\left( 2, 2 \right)\]
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Midpoint Formula
The midpoint of a line segment between two points (x₁, y₁) and (x₂, y₂) is found by averaging their coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2). This gives the exact center point of the segment.
Recommended video:
6:36
Quadratic Formula
Coordinate Geometry
Coordinate geometry involves representing geometric figures using coordinates on the Cartesian plane. Understanding how points, lines, and segments are expressed in (x, y) form is essential for applying formulas like the midpoint.
Recommended video:
05:32Intro to Polar Coordinates
Application of Arithmetic Operations
Calculating the midpoint requires basic arithmetic operations such as addition and division. Being comfortable with these operations ensures accurate computation of the average coordinates.
Recommended video:
04:12Algebraic Operations on Vectors
Related Practice
Textbook Question
CONCEPT PREVIEW Perform the indicated operation, and write each answer in lowest terms 3 7 —— + —— x x
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Textbook Question
CONCEPT PREVIEW Use choices A–D to answer each question.
A. 3x² - 17x - 6 = 0
B.(2x + 5)² = 7
C. x² + x = 12
D. (3x - 1) (x - 7) = 0
Which quadratic equation is set up for direct use of the square root property? Solve it.
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Textbook Question
CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. The point (-1, 3) lies in quadrant ________ in the rectangular coordinate system.
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Textbook Question
CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. The circle with equation x² + y² = 49 has center with coordinates ________ and radius equal to _______.
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Textbook Question
CONCEPT PREVIEW Evaluate each expression. 10³