Connecting algebra and geometry through coordinates

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1. Distance and midpoint in the coordinate plane

Warm-up

Three quick questions from earlier lessons.

Discover

From (1,1)(1, 1) to (4,5)(4, 5) you move 3 right and 4 up. How long is the straight segment?

  1. 5
  2. 7

Put A at (0, 0) and B at (4, 6). Where is M?

  1. (2, 3)
  2. (4, 6)

Key idea

Pythagoras on the grid, and averaging for the middle.

Distance: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. Midpoint: (x1+x22,y1+y22)\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right).

Worked example

A(−2,3)A(-2, 3), B(4,−5)B(4, -5).

d=36+64=10d = \sqrt{36 + 64} = 10
Δx = 6, Δy = −8.
Midpoint (1,−1)(1, -1)
Average the coordinates.

Your turn, guided

Your turn: the distance from (0,0)(0, 0) to (5,12)(5, 12).

1313
√(25 + 144).

Practice ★

  1. Find the distance between (−1,−3)(-1, -3) and (2,1)(2, 1).
  2. Find the distance between (−3,3)(-3, 3) and (0,7)(0, 7).
  1. Tap the midpoint of the segment from (−6,−5)(-6, -5) to (−4,−1)(-4, -1).
  2. Tap the midpoint of the segment from (−7,−8)(-7, -8) to (1,−4)(1, -4).

Watch out!

Practice ★★

  1. Find the exact distance between (0,−2)(0, -2) and (1,−4)(1, -4).
  2. Find the exact distance between (2,−2)(2, -2) and (3,−6)(3, -6).

Exit ticket

Three questions to check you have got it.

  1. Find the distance between (−1,−3)(-1, -3) and (2,1)(2, 1).
  2. Find the distance between (−3,3)(-3, 3) and (0,7)(0, 7).
  3. Find the distance between (2,5)(2, 5) and (11,17)(11, 17).

Summary card

Distance and midpoint

d=Δx2+Δy2d = \sqrt{\Delta x^2 + \Delta y^2}; midpoint = average of the coordinates.

Watch out: average, not half the difference.

2. Slopes of parallel and perpendicular lines

Warm-up

Three quick questions from earlier lessons.

Discover

Rotate a line of slope 2 by 90∘90^\circ. Its "1 right, 2 up" becomes "2 left, 1 up". What is the new slope?

  1. −12-\tfrac{1}{2}
  2. 12\tfrac{1}{2}

Key idea

Slopes tell parallel and perpendicular apart.

Slope m=y2−y1x2−x1m = \dfrac{y_2 - y_1}{x_2 - x_1}. Parallel lines have equal slopes. Perpendicular lines (neither vertical) have slopes with product −1-1: m2=−1m1m_2 = -\dfrac{1}{m_1}.

Worked example

Line through (1,2)(1, 2) and (5,4)(5, 4).

m=12m = \tfrac{1}{2}
2/4.
Parallel: 12\tfrac{1}{2}
Same slope.
Perpendicular: −2-2
Negative reciprocal.

Your turn, guided

Your turn: perpendicular to slope 34\tfrac{3}{4}.

−43-\tfrac{4}{3}
Flip and change sign.

Practice ★

  1. A line has slope 1/81/8. Find the slope of a line perpendicular to it.
  2. A line has slope 8/48/4. Find the slope of a line perpendicular to it.
  1. Are the lines y=6x+1y = 6x + 1 and y=−16x+5y = -\tfrac{1}{ 6 }x + 5 parallel, perpendicular or neither?
    1. Perpendicular: the product of the slopes is −1
    2. Parallel
    3. Neither

Watch out!

Practice ★★

  1. Find the slope of the line through (0,5)(0, 5) and (−5,−3)(-5, -3).
  2. Find the slope of the line through (−5,−2)(-5, -2) and (5,4)(5, 4).

Exit ticket

Three questions to check you have got it.

  1. A line has slope 1/81/8. Find the slope of a line perpendicular to it.
  2. A line has slope 8/48/4. Find the slope of a line perpendicular to it.
  3. A line has slope 4/74/7. Find the slope of a line perpendicular to it.

Summary card

Parallel and perpendicular slopes

Parallel: equal. Perpendicular: m1m2=−1m_1 m_2 = -1.

Watch out: flip and change the sign.

3. Equations of lines through a point

Warm-up

Three quick questions from earlier lessons.

Discover

A line has slope 2 and passes through (3,1)(3, 1). Does it pass through (4,3)(4, 3)?

  1. Yes
  2. No

Key idea

A point and a slope fix a line.

Point-slope form: y=y1+m(x−x1)y = y_1 + m(x - x_1). To get y=mx+by = mx + b, substitute the point to find bb. A line parallel (perpendicular) to a given line uses the same (negative reciprocal) slope, with its own intercept.

Worked example

Through (2,5)(2, 5), parallel to y=3x−4y = 3x - 4.

m=3m = 3
Same slope.
b=−1b = -1
5 = 3·2 + b.
y=3x−1y = 3x - 1
The line.

Your turn, guided

Your turn: slope −2-2 through (1,4)(1, 4). The intercept bb.

b=6b = 6
4 = −2 + b.

Practice ★

  1. Write the line with slope 33 through (4,−1)(4, -1): enter the right-hand side of y=…y = \ldots
  2. Write the line with slope 11 through (2,−1)(2, -1): enter the right-hand side of y=…y = \ldots
  1. Find the yy-intercept of the line parallel to y=−3x+−1y = -3x + -1 through (−3,−5)(-3, -5).
  2. Find the yy-intercept of the line parallel to y=1x+−1y = 1x + -1 through (−1,2)(-1, 2).

Watch out!

Practice ★★

  1. Find the yy-intercept of the line perpendicular to y=4xy = 4x through (−3,−3)(-3, -3).
  1. Boss problem. A(−2,−2)A(-2, -2) and B(2,0)B(2, 0). Find the perpendicular bisector of ABAB.

Exit ticket

Three questions to check you have got it.

  1. Write the line with slope 33 through (4,−1)(4, -1): enter the right-hand side of y=…y = \ldots
  2. Write the line with slope 11 through (2,−1)(2, -1): enter the right-hand side of y=…y = \ldots
  3. Write the line with slope 55 through (4,−1)(4, -1): enter the right-hand side of y=…y = \ldots

Summary card

Equations of lines

y=y1+m(x−x1)y = y_1 + m(x - x_1); find bb by substituting the point.

Watch out: parallel lines share the slope, not the intercept.

Answers: Connecting algebra and geometry through coordinates

1. Distance and midpoint in the coordinate plane

A

  1. 55
  2. 55

B

  1. (−5,−3)(-5, -3)
  2. (−3,−6)(-3, -6)

C

  1. 5\sqrt{5}
  2. 17\sqrt{17}

D

  1. 55
  2. 55
  3. 1515

2. Slopes of parallel and perpendicular lines

A

  1. −8-8
  2. −12- \frac{1}{2}

B

  1. Perpendicular: the product of the slopes is −1

C

  1. 85\frac{8}{5}
  2. 35\frac{3}{5}

D

  1. −8-8
  2. −12- \frac{1}{2}
  3. −74- \frac{7}{4}

3. Equations of lines through a point

A

  1. 3x−133 x - 13
  2. x−3x - 3

B

  1. −14-14
  2. 33

C

  1. −154- \frac{15}{4}

D

  1. −2-2

E

  1. 3x−133 x - 13
  2. x−3x - 3
  3. 5x−215 x - 21