Find and Use Slopes of Lines Write and Graph Equations of Lines Objectives: To find the slopes of lines To find the slopes of parallel and perpendicular lines To graph and write equations based on the Slope-Intercept Form, Standard Form, or Point-Slope Form of a Line

Slope Summary Summarize your findings about slope in the table below: m = undef Insert Picture As the absolute value of the slope of a line increases, --?--. the line gets steeper.

Slope of a Line The slope of a line (or segment) through P1 and P2 with coordinates (x1,y1) and (x2,y2) where x1x2 is ryse

Example 2 Find the slope of the line containing the given points. Then describe the line as rising, falling, horizontal, or vertical. (6, −9) and (−3, −9) (8, 2) and (8, −5) (−1, 5) and (3, 3) (−2, −2) and (−1, 5)

Example 3 A line through points (5, -3) and (−4, y) has a slope of −1. Find the value of y.

Parallel and Perpendicular Two lines are parallel lines iff they have the same slope. Two lines are perpendicular lines iff their slopes are negative reciprocals.

Example 4 Tell whether the pair of lines are parallel, perpendicular, or neither Line 1: through (−2, 1) and (0, −5) Line 2: through (0, 1) and (−3, 10) Line 1: through (−2, 2) and (0, −1) Line 2: through (−4, −1) and (2, 3)

Example 5 Line k passes through (0, 3) and (5, 2). Graph the line perpendicular to k that passes through point (1, 2).

Example 6 Find the value of y so that the line passing through the points (3, y) and (−5, −6) is perpendicular to the line that passes through the points (−2, −7) and (10, 1).

Example 7 Find the value of k so that the line through the points (k – 3, k + 2) and (2, 1) is parallel to the line through the points (−1, 1) and (3, 9).

Tangent

Tangent A line is a tangent if and only if it intersects a circle in one point.

Tangent Line Theorem In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.

Example 8 The center of a circle has coordinates (1, 2). The point (3, -1) lies on this circle. Find the slope of the tangent line at (3, -1).

Intercepts The x-intercept of a graph is where it intersects the x-axis. (a, 0) The y-intercept of a graph is where it intersects the y-axis. (0, b)

Equation of a Horizontal Line Equation of a Vertical Line Slope-Intercept Slope-Intercept Form of a Line: If the graph of a line has slope m and a y-intercept of (0, b), then the equation of the line can be written in the form y = mx + b. Equation of a Horizontal Line Equation of a Vertical Line y = b (where b is the y-intercept) x = a (where a is the x-intercept)

Example 9 Find the equation of the line with the set of solutions shown in the table. x 1 3 5 7 9 … y 11 17 23 29

Example 10 Graph the equation:

Slope-Intercept To graph an equation in slope-intercept form: Solve for y to put into slope-intercept form. Plot the y-intercept (0, b). Use the slope m to plot a second point. Connect the dots.

Example 11 Graph the equation:

Standard Form Standard Form of a Line The standard form of a linear equation is Ax + By = C, where A and B are not both zero. A, B, and C are usually integers.

Standard Form To graph an equation in standard form: Write equation in standard form. Let x = 0 and solve for y. This is your y-intercept. Let y = 0 and solve for x. This is your x-intercept. Connect the dots.

Example 12 Without your graphing calculator, graph each of the following: y = −x + 2 y = (2/5)x + 4 f (x) = 1 – 3x 8y = −2x + 20

Example 13 Graph each of the following: x = 1 y = −4

Example 14 A line has a slope of −3 and a y-intercept of (0, 5). Write the equation of the line.

Example 15 A line has a slope of ½ and contains the point (8, −9). Write the equation of the line.

Point-Slope Form Given the slope and a point on a line, you could easily find the equation using the slope-intercept form. Alternatively, you could use the point-slope form of a line. Point-Slope Form of a Line: A line through (x1, y1) with slope m can be written in the form y – y1 = m(x – x1).

Example 16 Find the equation of the line that contains the points (−2, 5) and (1, 2).

Example 17 Write the equation of the line shown in the graph.

Example 18 Write an equation of the line that passes through the point (−2, 1) and is: Parallel to the line y = −3x + 1 Perpendicular to the line y = −3x + 1

Example 19 Find the equation of the perpendicular bisector of the segment with endpoints (-4, 3) and (8, -1).

Example 20 The center of a circle has coordinates (1, 2). The point (3, −1) lies on this circle. Find the equation of the tangent line at (3, −1).

Distance from a Point to a Line The distance from a point to a line is the length of the perpendicular segment from the point to the line. This is the shortest distance from the point to the line.

Distance Between Parallel Lines Likewise, the distance between two parallel lines is the length of any perpendicular segment joining the two lines.

Prove Theorems About Perpendicular Lines 3.6 Proving Theorems about Perpendicular Lines Prove Theorems About Perpendicular Lines Objectives: To prove theorems about perpendicular lines

The Proof Game! Here’s your chance to play the game that is quickly becoming a favorite among America’s teenagers: The Proof Game! In this game, your group will be given one theorem. You are to collaborate with your group members to prove this theorem. Then you must choose someone to present said proof to the class. That presenter will earn a delicious reward!

Theorems Galore! If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. Proof Hints: Write an equation based on the angles forming a linear pair. Do some substitution and solve for one of the angles. It should be 90°.

Theorems Galore! If two lines are perpendicular, then they intersect to form four right angles. Proof Hints: Use definition of perpendicular lines to find one right angle. Use vertical and linear pairs of angles to find three more right angles.

Theorems Galore! If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. Proof Hints: Use definition of perpendicular to get the measure of ABC. Use Angle Addition Postulate, Substitution, and Definition of complementary angles to finish the proof.

Theorems Galore! Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Proof Hints: Use definition of perpendicular lines to find one right angle. Use Corresponding Angles Postulate to find a right angle on the other line. j

Theorems Galore! Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Proof Hints: Use definition of perpendicular lines to find a right angle on each parallel line. Use Converse of Corresponding Angles Postulate to prove the lines are parallel.