Chin-Sung Lin

Mr. Chin-Sung Lin  Distance Formula  Midpoint Formula  Slope Formula  Parallel Lines  Perpendicular Lines

Mr. Chin-Sung Lin

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

Calculate the distance between A ( 4, 5 ) and B ( 1, 1 ) Mr. Chin-Sung Lin

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

Calculate the distance between A ( 9, 5 ) and B ( 1, 5 ) Mr. Chin-Sung Lin

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 M = ( ) A (x 1, y 1 )B (x 2, y 2 )M (x, y) Mr. Chin-Sung Lin

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

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

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 x 2 - x 1 m = A (x 1, y 1 ) B (x 2, y 2 ) Mr. Chin-Sung Lin

Calculate the slope of AB, where A ( 4, 5 ) and B ( 2, 1 ) Mr. Chin-Sung Lin

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

Positive slope Mr. Chin-Sung Lin

Negative slope Mr. Chin-Sung Lin

Zero slope

Undefined slope Mr. Chin-Sung Lin

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

If AB is parallel to CD where A ( 2, 3 ) and B ( 4, 9 ), calculate the slope of CD Mr. Chin-Sung Lin

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

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

If AB is perpendicular to CD where A ( 1, 2 ) and B ( 3, 6 ), calculate the slope of CD Mr. Chin-Sung Lin

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

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

Linear equation can be written in slope- intercept form: y = mx + b where m is the slope b is the y-intercept slope: m b Mr. Chin-Sung Lin

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

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 = 3x + 2 Mr. Chin-Sung Lin

Given: y-intercept b and a point (x 1, y 1 ) (0, b) (x 1, y 1 ) Mr. Chin-Sung Lin

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 Step 3: Write the equation y = mx + b (0, b) (x 1, y 1 ) Mr. Chin-Sung Lin

Given: Two points (0, 4) and (2, 0) (0, 4) (2, 0) Mr. Chin-Sung Lin

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 = -2x + 4 (0, 4) (2, 0) Mr. Chin-Sung Lin

A line passing through (2, 3) and the y-intercept is -5. Write the equation Mr. Chin-Sung Lin

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

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

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

Given: Two points (x 1, y 1 ) and (x 2, y 2 ) (x 1, y 1 ) (x 2, y 2 ) Mr. Chin-Sung Lin

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 ) Step 2: Write the equation using slope m and any point y – y 1 = m(x – x 1 ) (x 1, y 1 ) (x 2, y 2 ) Mr. Chin-Sung Lin

Given: Two points (3, 1) and (1, 4) (1, 4) (3, 1) Mr. Chin-Sung Lin

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

Given: Two points (-2, 7) and (2, 3) Mr. Chin-Sung Lin

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

Write an equation of the line passing through the point (- 1, 1) that is parallel to the line y = 2x - 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 = 2x + 3 Mr. Chin-Sung Lin

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

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

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

Write an equation of the line passing through the point (2, 3) that is perpendicular to the line y = -2x + 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

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

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

Based on the information in the graph, write the equations of line P and line Q in both slope- intercept form and point-slope form Mr. Chin-Sung Lin (4, 3) y = 2x K P Q

Mr. Chin-Sung Lin

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

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

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

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

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

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

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

Vertices definition in coordinate geometry: Any triangle— (a, 0), (0, b), (c, 0) Mr. Chin-Sung Lin (0, b) (a, 0) (c, 0) (0, b) (a, 0) (c, 0)

Vertices definition in coordinate geometry: Right triangle— (a, 0), (0, b), (0, 0) Mr. Chin-Sung Lin (0, b) (0, 0) (a, 0)

Vertices definition in coordinate geometry: Isosceles triangle— (-a, 0), (0, b), (a, 0) Mr. Chin-Sung Lin (0, b) (-a, 0) (a, 0)

Vertices definition in coordinate geometry: Midpoint of segments— (2a, 0), (0, 2b), (2c, 0) Mr. Chin-Sung Lin (0, 2b) (2a, 0) (2c, 0)

Prove that the midpoint of the hypotenuse of a right triangle is equidistance from the vertices Mr. Chin-Sung Lin

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

Prove that the midpoint of the hypotenuse of a right triangle is equidistance from the vertices Mr. Chin-Sung Lin B (0, 2b) A (2a, 0) C(0, 0) M

Mr. Chin-Sung Lin

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

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

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

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

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

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

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

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

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

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