2.3 Apply slope, midpoint, length - Course Webpage: … · midpoint of a line segment o Understand...
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MPM2D Date: ________________________________ 2.3 Apply Slope, Midpoint, and Length Formulas Learning Goals Success Criteria Solve problems involving slope, length and midpoint of a line segment o Understand the difference between slope, length, and midpoint o Determine the equation for a right bisector of a line o Determine the shortest distance from a point to a line Recall: Line segment with endpoints ! , ! and ! , ! Slope Midpoint Length = ! − ! ! − ! ! + ! 2 , ! + ! 2 = ! − ! ! + ! − ! ! Two lines are parallel if their slopes are the same Two lines are perpendicular if their slopes are negative reciprocals ! = − ! ! Example 1: The vertices of Δ are A(5, 5), B(-3, -1) and C(1, -3). Determine whether Δ is a right triangle. Step 1: Draw the Triangle Step 2: Determine the three (3) slopes !" = = −1 − 5 −3 − 5 = −6 −8 = 3 4 !" = −3 − −1 1 − −3 = − 2 4 = − 1 2 !" = −3 − 5 1 − 5 = −8 −4 = 2 Right, ⏊, 90°, perpendicular Step 3: Compare slopes (Are any negative reciprocals?) !" = − ! ! !" so yes Δ is a right triangle
Transcript of 2.3 Apply slope, midpoint, length - Course Webpage: … · midpoint of a line segment o Understand...
MPM2D Date: ________________________________
2.3 Apply Slope, Midpoint, and Length Formulas
Learning Goals Success Criteria
Solve problems involving slope, length and midpoint of a line segment
o Understand the difference between slope, length, and midpoint
o Determine the equation for a right bisector of a line
o Determine the shortest distance from a point to a line
Recall: Line segment with endpoints 𝐴 𝑥!,𝑦! and 𝐵 𝑥!,𝑦!
Slope Midpoint Length 𝑚 =
𝑦! − 𝑦!𝑥! − 𝑥!
𝑀𝑥! + 𝑥!2 ,
𝑦! + 𝑦!2 𝑑 = 𝑥! − 𝑥! ! + 𝑦! − 𝑦! !
Two lines are parallel if their slopes are the same Two lines are perpendicular if their slopes are negative reciprocals 𝑚! = − !
!
Example 1: The vertices of Δ𝐴𝐵𝐶 are A(5, 5), B(-3, -1) and C(1, -3). Determine whether Δ𝐴𝐵𝐶 is a right triangle. Step 1: Draw the Triangle Step 2: Determine the three (3) slopes
𝑚!" =𝑟𝑖𝑠𝑒𝑟𝑢𝑛
=−1− 5−3− 5
=−6−8
=34
𝑚!" =−3− −11− −3
= −24
= −12
𝑚!" =−3− 51− 5
=−8−4
= 2
Right, ⏊, 90°, perpendicular
Step 3: Compare slopes (Are any negative reciprocals?) 𝑚!" = − !
!!"
so yes Δ𝐴𝐵𝐶 is a right triangle
0 1 2 3 4 5 6 7 8 9 10 11
1
2
3
4
5
6
7
A
B
C
Example 2: Chloe and Chloe are walking in the forest and they have reached the point that has coordinates C(6,1). They want to reach the straight road that connects the points A(2,2) and B(10,6). Chloe and Chloe want to walk the shortest distance possible back to the road. Each square represents 500 m.
(a) Determine the equation of the line for the road.
(b) Determine the equation of the line Chloe and Chloe should walk on.
(c) At what point will they reach the road? (d) Determine the distance of this path
Chloe and Chloe must take, to the nearest metre.
(a) Equation of a line 𝑦 = 𝑚𝑥 + 𝑏
𝑚 =6− 210− 2
=48
=12
Use point A: 𝑥 = 2,𝑦 = 2
2 =12 2 + 𝑏
𝑏 = 2− 1 𝑏 = 1 Therefore, the equation of the line for the road is 𝒚 = 𝟏
𝟐𝒙+ 𝟏
(b) Slope of shortest distance is perpendicular to AB 𝑚!"#$ =
!!
so 𝑚!"#! = −2 Point C(6, 1) is on the line so … 𝑦 = 𝑚𝑥 + 𝑏 1 = −2 6 + 𝑏 𝑏 = 13 Therefore the equation of the line that Chloe and Chloe should walk in is 𝒚 = −𝟐𝒙+ 𝟏
(c) Find intersection of 𝑦 = !!𝑥 + 1 and 𝑦 = −2𝑥 + 13
Set (1) = (2) !!𝑥 + 1 = −2𝑥 + 13
12 𝑥 + 2𝑥 = 13− 1
!!𝑥 = 12
𝑥 = !"!
= 4.8
Sub 𝑥 = 4.8 into (2) 𝑦 = −2 4.8 + 13 = 3.4 They will reach the road at (4.8, 3.4)
(d) Find distance between C(6,1) and D(4.8, 3.4)𝑑 = 6− 4.8 ! + 1− 3.4 ! = 2.8 ! + −2.4 ! = 13.6 ≈ 3.7
1 square = 500 m 3.7 squares = 1843.9 The distance of the path Chloe and Chloe must take is 1844 m.
