1 SLOPE FORMULA SLOPE FORMULA: PROBLEMS SLOPE: RUN VS RISE. PROBLEMS FALLING OR RISING LINES? SLOPE CASES PARALLEL VS PERPENDICULAR PARALLEL VS SKEW AND PERPENDICULAR Standards 7 and 17 END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

2 STANDARD 7: Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and properties of circles. ESTÁNDAR 7: Los estudiantes prueban y usan teoremas involucrando las propiedades de líneas paralelas cortadas por una transversal, las propiedades de cuadriláteros, y las propiedades de círculos. STANDARD 17: Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles. ESTÁNDAR 17: Los estudiantes prueban teoremas usando geometría coordenada, incluyendo el punto medio de un segmento, la fórmula de la distancia y varias formas de ecuaciones de líneas y círculos. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3 Standard 17 m= y 1 y 2 x 1 x Segment AB:A(3,2), B(7,5) y 1 y 2 x 1 x = = 3 4 Segment CD:C(8,2), D(10,1) y 1 y 2 x 1 x = 1 -2 m = - - AB m = - - CD Slope Formula: Find the slope for the segments at the right: x y A B C D The slope is the inclination of a line or segment. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4 m= y 1 y 2 x 1 x x y Using the slope formula below, find the slope for the lines in the coordinate plane: a b c d (-8, 7) (-4, -4) (-1, 1) (1, 5) (3, 2) (10, 2) (2, -5) (6, -8) (1,5) (-1,1) x 2 1 x 1 y 2 5 y 1 1 = = 4 2 m = - - a Line a:a: = =2 (3,2) (10,2) x 2 3 x 1 10 y 2 2 y 1 2 = 0 -7 m = - - b Line b:b: =0 (6,-8) (2,-5) x 2 6 x 1 2 y 2 -8 y 1 -5 = -3 4 m = - - c Line c:c: = = (-8,7) (-4,-4) x 2 -8 x 1 -4 y 2 7 y 1 = m = - - d Line d:d: = Standard 17 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5 m= y 1 y 2 x 1 x x y Using the run vs. the rise method, find the slope for the lines in the coordinate plane: a b c d = = m = a = m = b = m = c = m = d Line a:a: b:b: c:c: d:d: = rise run Standard 17 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6 x y c + - FALLING TO THE RIGHT When we move to the right from one point to the other, we go down the right and the slope is negative! - + m = c Slope for c:c: = - Standard 17 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7 Standard 5 x y a m = a =+=+ Slope for a:a: FALLING TO THE LEFT When we move to the left from one point to the other, we go down and the slope is always positive! PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8 x y c - + RISING TO THE LEFT When we move to the left from one point to the other, we go up to the left and the slope is negative! + - m = c Slope for c:c: = - Standard 17 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9 x y a m = a =+=+ Slope for a:a: RISING TO THE RIGHT When we move to the right from one point to the other, we go up and the slope is always positive! Standard 17 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10 x y b m = b =0=0 Slope for b:b: Slope of a horizontal line is always 0! x y a m = a =+=+ Slope for a:a: Slope of a line that falls to the left is always POSITIVE! x y c m = c Slope for c:c: = - The slope of a line that falls to the right is always NEGATIVE! x y + 0 m = d Slope for d:d: = not defined! The slope of a vertical line IS NEVER DEFINED! SUMMARIZING FINDINGS d + Standard 17 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

x y Parallel Lines have the same slope! +2 m = a =1 +2 m = b =1 a b PARALLEL VS. PERPENDICULAR x y c d m = c =-1 +2 m = d =1 The slope product of perpendicular lines is -1! m = d m c (-1)(1) m = a m b d m c PARALLEL PERPENDICULAR =-1 Line a:a: b:b: c:c: d:d: Standard 17 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

12 m= y 1 y 2 x 1 x Segment AB:A(2,1), B(6,4) y 1 y 2 x 1 x = = 3 4 Segment BC:B(6,4), C(9,0) y 1 y 2 x 1 x = = = -1 m = - - AB m = - - BC Using the slope formula: Prove that segments AB and BC are perpendicular: x y A B C Is the product of the slopes -1? The product of the slopes is -1. So, They are perpendicular. Standard 17 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

13 Standard 7 PARALLEL VS SKEW D A B C E F GH Lines AB and DC are PARALLEL AB DC They are on the same plane and never cross PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14 Standard 7 PARALLEL VS SKEW D A B C E F GH Lines CG and BF are PARALLEL CG BF They are on the same plane and never cross May you name other lines that are parallel? AD HE AB EF DC HG EH FG PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

15 Standard 7 PARALLEL VS SKEW D A B C E F GH Lines AB and CG are SKEW They are on different planes and never cross PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

16 Standard 7 PARALLEL VS SKEW D A B C E F GH Lines DC and BF are SKEW They are on different planes and never cross May you name other lines that are skew? DC and AE HG and AE FG and AE DH and EF PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

17 Standard 7 D A B E F GH May you name two planes that are parallel? Planes ADH and BCG C PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

18 Standard 7 D A B E F GH May you name two planes that are parallel? Planes ADH and BCG C PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

19 Standard 7 D A B C E F GH May you name two planes that are parallel? Planes ABC and EFG PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

20 Standard 7 D A B C E F GH May you name two planes that are parallel? Planes ABC and EFG PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

21 Standard 7 D A B C E F GH May you name two planes that are parallel? Planes ABC and EFG Also: Planes AEF and DHG Planes ADH and BCG PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

22 Standard 7 PERPENDICULAR LINES D A B C E F GH Lines DC and CG are PERPENDICULAR They intersect forming a right angle lying on the same plane. May you name other lines that are perpendicular? DC CB FG CG HG EH DH AD PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

23 Standard 7 D A B C E F GH May you name two planes that are perpendicular? Planes DAB and CBF Also: Planes DAB and DHG Planes EHG and DAE PRESENTATION CREATED BY SIMON PEREZ. All rights reserved