Area of triangle? Area of square? Area of Rectangle? distance formula: slope point form: slope intercept form: February 22, 2017
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- Bertha Dorsey
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1 Formula Page for this Unit! Quiz tomorrow! Slope Formula: rise run slope intercept form: slope point form: distance formula: Area of triangle? Area of parallelogram? Area of square? Area of Rectangle? 1
2 1. Slope intercept form 2. Distance Formula 3. Slope formula 4. Midpoint Formula 5. Slope/point formula 6. Area of square 7. Area of triangle 8. Area of rectangle 9. Area of parallelogram 2
3 Module 2: Coordinate Proof Using Slope and Distance TN Ready Standards G GPE.B.4, 5, 6, & 7 B.4 Use coordinates to prove simple geometric theorems algebraically B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. 3
4 ' ; Theorem: Slope Criteria for Parallel Lines and Perpendicular Lines Two nonvertical lines are parallel if and only if they have the same slope. Two nonvertical lines are perpendicular if and only if the product of their slopes is 1. * We won't understand this until we know what slope means! What is the slope? 4
5 3 ways to find slope! 5
6 Pull There are 4 different kinds of slope. You need to know each by their distinct picture. **Remember you always read graphs from left to right. Positive Negative Horizontal line = zero slope Undefine Vertical line. 6
7 When given a graph find slope by counting rise / run from point to point on the graph. Pull 7
8 Find the slope of the line by counting (rise / run) opposite of rise is to fall. 8
9 Find the slope of the line by counting (rise / run). 9
10 When given an equation such as y=mx+b slope is the coefficient to x. (m) ALWAYS! Pull 10
11 What is the slope of the given equation (function)? y = 4x + 6 m = 4 y = 1/3x 9 m = 1/3 11
12 y = x + 3/4 m = 1 4x + 2y = 16 4x = 4x y = 4x y = 2x + 8 m = 2 12
13 2x + y = 8 2x = 2x + 8 y = 2x + 8 m = 2 y = 2 + 5x y = 5x 2 m = 5 13
14 Pull When given two points from the line, plug them into the slope formula and solve for slope. m = y 2 y 1 x 2 x 1 Find the slope of the line that contains (5, 2), ( 1, 4) 14
15 Find the slope of the line that contains the given points. Pull ( 2, 4), (2,4) (0,2), ( 2, 0) (3, 4), (4, 2) Pull Pull (0, 6), ( 1, 1) Pull (1, 9), (3,9) Pull (5, 7), (5, 1) Pull 15
16 RECAP! How do you find slope if you are given a graph? How do you find slope if you are given an equation? What might you have to do to the equation first? How do you find slope if given two points from the line? What do the little 2's & 1's mean in the slope formula? 16
17 So if we know what slope is now, how can we determine if two lines are parallel? or perpendicular? Two nonvertical lines are parallel if and only if they have the same slope. Two nonvertical lines are perpendicular if and only if the product of their slopes is 1. 17
18 What are Parallel and Perpendicular Lines? Draw an example of each in your notebook. 18
19 Let's take a look at a graph of 2 parallel lines. What do you notice? 19
20 Perpendicular Lines 20
21 21
22 Parallel Lines have the same. How in the world will we remember this? I KNOW!! 22
23 Parallel Lines have the same slope. Parallel Lines have the same slope. Parallel Lines have the same slope, and never will they meet. (to the tune of Skip to My Lou) 23
24 What about perpendicular lines? Slopes of perpendicular lines have NEGATIVE RECIPROCAL slopes. Which means... take the slope and flip it, then make it opposite. m = (1/2) What is the perpendicular slope? 24
25 Perpen, Perpendicular. Right Angle. Perpen, Perpendicular. Right Angle. First you take the slope and you flip it, you flip it. Then you take the slope and make it opposite, opposite. Perpen, Perpendicular. Right Angle. (to the tune of Peanut Butter and Jelly) 25
26 Write the equation of a line that is PARALLEL to the following: y=4x+2 y=-2/3x-1 26
27 Write the equation of a line that is PERPENDICULAR to the following: y=4x+2 y=-2/3x-1 27
28 Write an equation of a line that is parallel to the following and goes through the following point: y=2/3x + 5 through the point (6,5) 28
29 Wait, how can we do that? Point/Slope form: go back 29
30 Write the equation of the line that is perpendicular to the line with the equation below: y = (1/3) x + 7 and goes through the point (2,3) 30
31 ACT STARTER: What is the least common multiple of 8, 16, 24 and 48? 31
32 Classifying Quadrilaterals 32
33 Special Quadrilaterals: Parallelogram A B Properties: A quadrilateral with both pairs of opposite sides parallel. Opposites sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary D C Diagonals bisect each other (Click on each property to have it fly in for effect.) 33
34 Special Quadrilaterals: Rectangle J K Properties: A parallelogram with four right angles. Has all the properties of a parallelogram. M L Diagonals are congruent. If only one right angle is marked, the figure is a rectangle. (Click on each property to have it fly in for effect.) 34
35 Special Quadrilaterals: Rhombus Properties: E A parallelogram with four congruent sides. All properties of a parallelogram apply. H F Each diagonal bisects two angles. Diagonals are perpendicular. G (Click on each property to have it fly in for effect.) 35
36 Special Quadrilaterals: Trapezoid Properties: A quadrilateral with exactly one pair of parallel sides. R S The parallel sides are called bases; the nonparallel sides are called legs. The base and one of the legs form the base angles. If legs are congruent, then it is an isosceles trapezoid. U Diagonals of an isosceles trapezoid are congruent. Both pairs of base angles of an isosceles trapezoid are congruent. T (Click on each property to have it fly in for effect.) 36
37 Special Quadrilaterals: Square N O Properties: A parallelogram with four congruent sides and four right angles. Has all the properties of a parallelogram, a rhombus and a rectangle. Diagonals are congruent and perpendicular. Q Diagonals bisect opposite angles. Diagonals bisect each other. P (Click on each property to have it fly in for effect.) 37
38 Special Quadrilaterals: Kite K Properties: A quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent. E I Diagonals are perpendicular. T (Click on each property to have it fly in for effect.) 38
39 Name the Quadrilateral Use the eraser to reveal the name of the quadrilateral found to the right of the figure. Rectangle Isosceles Trapezoid Parallelogram Label 39
40 Properties of Quadrilaterals Guess the correct answer and pop a balloon. rhombus square trapezoid kite 40
41 41
42 If you were given the coordinates of a figure and asked to classify what type of quadrilateral it is, what might you need to know besides slope? 42
43 What is the distance formula and midpoint formula? Pull 43
44 Applying Distance and Midpoint Formulas The distance d between any two points (x 1, y 1 ) and (x 2, y 2 ) in a coordinate plane is: (x 2, y 2 ) 44
45 Ex. 1 Find the distance between ( 1, 3) and (5, 2). Let (x 1, y 1 ) = ( 1, 3) and (x 2, y 2 ) = (5, 2) It doesn't matter which ordered pair is (x 1, y 1 ) and (x 2, y 2 ) units This means that the distance between the points is 45
46 Ex. 1 Find the distance between ( 1, 3) and (5, 2). Let (x 1, y 1 ) = ( 1, 3) and (x 2, y 2 ) = (5, 2) It doesn't matter which ordered pair is (x 1, y 1 ) and (x 2, y 2 ) units This means that the distance between the points is 46
47 47
48 48
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