Chap 8 Slopes and Equations of Lines Eleanor

Chap 8 Slopes and Equations of Lines Eleanor Roosevelt High School Geometry Chin-Sung Lin

ERHS Math Geometry Basic Geometry Formulas q Distance Formula q Midpoint Formula q Slope Formula Ø Parallel Lines Ø Perpendicular Lines Mr. Chin-Sung Lin

ERHS Math Geometry Distance Formula Mr. Chin-Sung Lin

ERHS Math Geometry Distance Formula Distance between two points A (x 1, y 1) and B (x 2, y 2) is given by distance formula d(A, B) =√(x 2 − x 1 )2 + (y 2 − y 1 )2 A (x 1, y 1) B (x 2, y 2) Mr. Chin-Sung Lin

ERHS Math Geometry Distance Formula - Example Calculate the distance between A (4, 5) and B (1, 1) Mr. Chin-Sung Lin

ERHS Math Geometry Distance Formula - Example Calculate the length of AB if the coordinates of A and B are (4, 15) and (-1, 3) respectively Mr. Chin-Sung Lin

ERHS Math Geometry Distance Formula - Example Calculate the distance between A (9, 5) and B (1, 5) Mr. Chin-Sung Lin

ERHS Math Geometry Midpoint Formula Mr. Chin-Sung Lin

ERHS Math Geometry Midpoint Formula If the coordinates of A and B are ( x 1, y 1) and ( x 2, y 2) respectively, then the midpoint, M, of AB is given by the midpoint formula x 1 + x 2, y 1+ y 2 M=( A (x 1, y 1) 2 M (x, y) 2 ) B (x 2, y 2) Mr. Chin-Sung Lin

ERHS Math Geometry Midpoint Formula - Example Calculate the midpoint of AB if the coordinates of A and B are (2, 7) and ( -6, 5) respectively Mr. Chin-Sung Lin

ERHS Math Geometry Midpoint Formula - Example M(1, -2) is the midpoint of AB and the coordinates of A are (-3, 2). Find the coordinates of B Mr. Chin-Sung Lin

ERHS Math Geometry Slope Formula Mr. Chin-Sung Lin

ERHS Math Geometry Slope Formula If the coordinates of A and B are (x 1, y 1) and (x 2, y 2) respectively, then the slope, m, of AB is given by the slope formula y 2 - y 1 y 2) , m= (x 2 B x 2 - x 1 A y 1) , (x 1 Mr. Chin-Sung Lin

ERHS Math Geometry Slope Formula - Example Calculate the slope of AB, where A (4, 5) and B (2, 1) Mr. Chin-Sung Lin

ERHS Math Geometry Slope Formula - Example Calculate the slope of AB, where A (4, 5) and B (2, 1) 5 -1 m= 4 -2 = 2 Mr. Chin-Sung Lin

ERHS Math Geometry Slope of Lines in the Coordinate Planes Positive slope Mr. Chin-Sung Lin

ERHS Math Geometry Slope of Lines in the Coordinate Planes Negative slope Mr. Chin-Sung Lin

ERHS Math Geometry Slope of Lines in the Coordinate Planes Zero slope Mr. Chin-Sung Lin

ERHS Math Geometry Slope of Lines in the Coordinate Planes Undefined slope Mr. Chin-Sung Lin

ERHS Math Geometry Slope and Parallel Lines The straight lines with slopes (m) and (n) are parallel to each other m n if and only if m = n Mr. Chin-Sung Lin

ERHS Math Geometry Slope and Parallel Lines - Example If AB is parallel to CD where A (2, 3) and B (4, 9), calculate the slope of CD Mr. Chin-Sung Lin

ERHS Math Geometry Slope and Parallel Lines - Example If AB is parallel to CD where A (2, 3) and B (4, 9), calculate the slope of CD m=n= 9 -3 4 -2 = 3 Mr. Chin-Sung Lin

ERHS Math Geometry Slope and Perpendicular Lines The straight lines with slopes (m) and (n) are mutually perpendicular n if and only if m · n = -1 m Mr. Chin-Sung Lin

ERHS Math Geometry Slope and Perpendicular Lines - Example If AB is perpendicular to CD where A (1, 2) and B (3, 6), calculate the slope of CD Mr. Chin-Sung Lin

ERHS Math Geometry Slope and Perpendicular Lines - Example If AB is perpendicular to CD where A (1, 2) and B (3, 6), calculate the slope of CD 6 -2 m= 3 -1 = 2 since m · n = -1, 2 · n = -1, so, n = -1/2 Mr. Chin-Sung Lin

ERHS Math Geometry Group Work Mr. Chin-Sung Lin

ERHS Math Geometry Parallel and Perpendicular Lines There are four points A (2, 6), B(6, 4), C(4, 0) and D(0, 2) on the coordinate plane. Identify the pairs of parallel and perpendicular lines Mr. Chin-Sung Lin

ERHS Math Geometry Equations of Lines Mr. Chin-Sung Lin

ERHS Math Geometry Slope Intercept Form Linear equation can be written in slopeintercept form: y = mx + b where m b is the slope is the y-intercept b slope: m Mr. Chin-Sung Lin

ERHS Math Geometry Write Slope Intercept Form Given: If the slope of a line is 3 and it passes through(0, 2), write the equation of the line in slope-intercept form Mr. Chin-Sung Lin

ERHS Math Geometry Write Slope Intercept Form Given: If the slope of a line is 3 and it passes through(0, 2), write the equation of the line in slope-intercept form m = 3, b = 2 y = 3 x + 2 Mr. Chin-Sung Lin

ERHS Math Geometry Write Slope Intercept Form Given: y-intercept b and a point (x 1, y 1) (0, b) (x 1, y 1) Mr. Chin-Sung Lin

ERHS Math Geometry Write Slope Intercept Form Given: y-intercept b and a point (x 1, y 1) Step 1: Find the slope m by choosing two points (0, b) and (x 1, y 1) on the graph of the line Step 2: Find the y-intercept b (0, b) Step 3: Write the equation y = mx + b (x 1, y 1) Mr. Chin-Sung Lin

ERHS Math Geometry Write Slope Intercept Form Given: Two points (0, 4) and (2, 0) (0, 4) (2, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Write Slope Intercept Form Given: Two points (0, 4) and (2, 0) Step 1: Find the slope by choosing two points on the graph of the line: m = (0 -4)/(2 -0) = -2 Step 2: Find the y-intercept: b = 4 Step 3: Write the equation: y = -2 x + 4 (0, 4) (2, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Write Slope Intercept Form - Example A line passing through (2, 3) and the y-intercept is -5. Write the equation Mr. Chin-Sung Lin

ERHS Math Geometry Point-Slope Form Linear equation can be written in point-slope form: y – y 1 = m(x – x 1) where m (x 1, y 1) is the slope is a point on the line (x 1, y 1) slope: m Mr. Chin-Sung Lin

ERHS Math Geometry Write Point-Slope Form Given: If the slope of a line is 3 and it passes through(5, 2), write the equation of the line in slope-intercept form Mr. Chin-Sung Lin

ERHS Math Geometry Write Point-Slope Form Given: If the slope of a line is 3 and it passes through(5, 2), write the equation of the line in slope-intercept form m = 3, (x 1, y 1) = (5, 2) y - 2 = 3(x – 5) Mr. Chin-Sung Lin

ERHS Math Geometry Write Point-Slope Form Given: Two points (x 1, y 1) and (x 2, y 2) (x 1, y 1) (x 2, y 2) Mr. Chin-Sung Lin

ERHS Math Geometry Write Point-Slope Form Given: Two points (x 1, y 1) and (x 2, y 2) Step 1: Find the slope m by plugging two points (x 1, y 1) and (x 2, y 2) into the slop formula m = (y 2 – y 1)/(x 2 – x 1) (x 1, y 1) Step 2: Write the equation using slope m and any point y – y 1 = m(x – x 1) (x 2, y 2) Mr. Chin-Sung Lin

ERHS Math Geometry Write Point-Slope Form Example Given: Two points (3, 1) and (1, 4) (3, 1) Mr. Chin-Sung Lin

ERHS Math Geometry Write Point-Slope Form Example Given: Two points (3, 1) and (1, 4) Step 1: Find the slope m by plugging two points (3, 1) and (1, 4) into the slop formula m = (4 – 1)/(1 – 3) = -3/2 Step 2: Write the equation y – 1 = (-3/2)(x – 3) (1, 4) (3, 1) Mr. Chin-Sung Lin

ERHS Math Geometry Write Point-Slope Form Example Given: Two points (-2, 7) and (2, 3) Mr. Chin-Sung Lin

ERHS Math Geometry Equations of Parallel & Perpendicular Lines Mr. Chin-Sung Lin

ERHS Math Geometry Equation of a Parallel Line Write an equation of the line passing through the point (-1, 1) that is parallel to the line y = 2 x – 3 Mr. Chin-Sung Lin

ERHS Math Geometry Equation of a Parallel Line Write an equation of the line passing through the point (1, 1) that is parallel to the line y = 2 x - 3 Step 1: Find the slope m from the given equation: since two lines are parallel, the slopes are the same, so: m = 2 Step 2: Find the y-intercept b by using the m = 2 and the given point (-1, 1): 1 = 2(-1) + b, so, b = 3 Step 3: Write the equation: y = 2 x + 3 Mr. Chin-Sung Lin

ERHS Math Geometry Equation of a Parallel Line - Example Write an equation of the line passing through the point (2, 3) that is parallel to the line y = x – 5 Mr. Chin-Sung Lin

ERHS Math Geometry Equation of a Parallel Line Write an equation of the line passing through the point (2, 0) that is parallel to the line y = x - 2 Mr. Chin-Sung Lin

ERHS Math Geometry Equation of a Perpendicular Line Write an equation of the line passing through the point (2, 3) that is perpendicular to the line y = -2 x + 2 Mr. Chin-Sung Lin

ERHS Math Geometry Equation of a Perpendicular Line Write an equation of the line passing through the point (2, 3) that is perpendicular to the line y = -2 x + 2 Step 1: Find the slope m from the given equation: since two lines are perpendicular, the product of the slopes is equal to -1, so: m = 1/2 Step 2: Find the y-intercept b by using the m = 2 and the given point (-1, 1): 3 = (1/2)(2) + b, so, b = 2 Step 3: Write the equation: y = (1/2)x + 2 Mr. Chin-Sung Lin

ERHS Math Geometry Equation of a Perpendicular Line Write an equation of the line passing through the point (1, 2) that is perpendicular to the line y = x + 3 Mr. Chin-Sung Lin

ERHS Math Geometry Equation of a Perpendicular Line Write an equation of the line passing through the point (4, 1) that is perpendicular to the line y = -x + 2 Mr. Chin-Sung Lin

ERHS Math Geometry Group Work Mr. Chin-Sung Lin

ERHS Math Geometry Equation of a Perpendicular Line Based on the information in the graph, write the equations of line P and line Q in both slopeintercept form and point-slope form Q K P y = 2 x -5 (4, 3) -2 Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof Two types of proofs in coordinate geometry: • Special cases Given ordered pairs of numbers, and prove something about a specific segment or polygon • General Theorems When the given information is a figure that represents a particular type of polygon, we must state the coordinates of its vertices in general terms using variables Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof Two skills of proofs in coordinate geometry: • Line segments bisect each other the midpoints of each segment are the same point • Two lines are perpendicular to each other the slope of one line is the negative reciprocal of the slope of the other Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof – Special Cases If the coordinates of four points are A(-3, 5), B(5, 1), C(-2, -3), and D(4, 9), prove that AB and CD are perpendicular bisector to each other Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof – Special Cases The vertices of rhombus ABCD are A(2, -3), B(5, 1), C(10, 1) and D(7, -3). (a) Prove that the diagonals bisect each other. (b) Prove that the diagonals are perpendicular to each other. Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof – Special Cases If the coordinates of three points are A(-1, 4), B(4, 7), and C(1, 2), prove that ABC is an isosceles triangle Mr. Chin-Sung Lin

ERHS Math Geometry Aim: Coordinate Proof Do. Now: If the coordinates of three points are A(-1, 4), B(4, 7), and C(1, 2), prove that ABC is an isosceles triangle Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof – Special Cases If the coordinates of three points are A(4, 3), B(6, 7), and C(-4, 7), prove that ΔABC is a right triangle. Which angle is the right angle? Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof – General Theorems Vertices definition in coordinate geometry: • Any triangle— (a, 0), (0, b), (c, 0) (0, b) (a, 0) (c, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof – General Theorems Vertices definition in coordinate geometry: • Right triangle— (a, 0), (0, b), (0, 0) (0, b) (0, 0) (a, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof – General Theorems Vertices definition in coordinate geometry: • Isosceles triangle— (-a, 0), (0, b), (a, 0) (0, b) (-a, 0) (a, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof – General Theorems Vertices definition in coordinate geometry: • Midpoint of segments— (2 a, 0), (0, 2 b), (2 c, 0) (0, 2 b) (2 a, 0) (2 c, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof – General Theorems Prove that the midpoint of the hypotenuse of a right triangle is equidistance from the vertices Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof – General Theorems Prove that the midpoint of the hypotenuse of a right triangle is equidistance from the vertices Given: Right triangle ABC whose vertices are A(2 a, 0), B(0, 2 b), and C(0, 0). Let M be the midpoint of the hypotenuse AB Prove: AM = BM = CM C(0, 0) B (0, 2 b) M A (2 a, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Coordinate Proof – General Theorems Prove that the midpoint of the hypotenuse of a right triangle is equidistance from the vertices B (0, 2 b) M C(0, 0) A (2 a, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Aim: Concurrence of the Altitudes of a Triangle Do. Now: Prove that the midpoint of the hypotenuse of a right triangle is equidistance from the vertices B (0, 2 b) M C(0, 0) A (2 a, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Concurrence of the Altitudes of a Triangle Mr. Chin-Sung Lin

ERHS Math Geometry Altitude Concurrence - Orthocenter: The altitudes of a triangle intersect in one point B(0, b) Acute Triangle A(a, 0) C(c, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Altitude Concurrence - Orthocenter Theorem: The altitudes of a triangle are concurrent (intersecting in one point— orthocenter) B(0, b) Acute Triangle A(a, 0) C(c, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Altitude Concurrence - Orthocenter: The altitudes of a triangle intersect in one point B(0, b) Right Triangle A(a, 0) C(c, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Altitude Concurrence - Orthocenter Theorem: The altitudes of a triangle are concurrent (intersecting in one point— orthocenter) B(0, b) Right Triangle A(a, 0) C(c, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Altitude Concurrence - Orthocenter: The altitudes of a triangle intersect in one point Obtuse Triangle B(0, b) A(a, 0) C(c, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Altitude Concurrence - Orthocenter Theorem: The altitudes of a triangle are concurrent (intersecting in one point— orthocenter) Obtuse Triangle B(0, b) A(a, 0) C(c, 0) Mr. Chin-Sung Lin

ERHS Math Geometry Orthocenter The coordinates of the vertices of ΔABC are A(0, 0), B(-2, 6), and C(4, 0). Find the coordinates of the orthocenter of the triangle Mr. Chin-Sung Lin

ERHS Math Geometry Orthocenter The coordinates of the vertices of ΔABC are A(0, 0), B(-2, 6), and C(4, 0). Find the coordinates of the orthocenter of the triangle Answer: (-2, -2) Mr. Chin-Sung Lin

ERHS Math Geometry Orthocenter The coordinates of the vertices of ΔABC are A(0, 0), B(3, 4), and C(2, 1). Find the coordinates of the orthocenter of the triangle Mr. Chin-Sung Lin

ERHS Math Geometry Orthocenter The coordinates of the vertices of ΔABC are A(0, 0), B(3, 4), and C(2, 1). Find the coordinates of the orthocenter of the triangle Answer: (6, -2) Mr. Chin-Sung Lin

ERHS Math Geometry Appendix Centroid & Circumcenter Mr. Chin-Sung Lin

ERHS Math Geometry Centroid The three medians (the lines drawn from the vertices to the bisectors of the opposite sides) meet in the centroid or center of mass (center of gravity). The centroid divides each median in a ratio of 2: 1 Mr. Chin-Sung Lin

ERHS Math Geometry Centroid The coordinates of the vertices of ΔABC are A(0, 0), B(4, 0), and C(0, 3). (a) Find the coordinates of the centroid of the triangle. (b) Prove that the centroid divides each median in a ratio of 2: 1 Mr. Chin-Sung Lin

ERHS Math Geometry Circumcenter The intersection of the 3 perpendicular bisectors of a triangle The circumcenter is equidistant to three vertices * perpendicular bisector - a line perpendicular to a side of a triangle passing through its midpoint Mr. Chin-Sung Lin

ERHS Math Geometry Circumcenter The coordinates of the vertices of ΔABC are A(0, 0), B(4, 0), and C(0, 3). (a) Find the coordinates of the circumcenter of the triangle. (b) Prove that the circumcenter is equidistant to three vertices Mr. Chin-Sung Lin

ERHS Math Geometry The End Mr. Chin-Sung Lin