Thirty-Six

Unit 6 · GeometryG5 · Coordinate Geometry · Aerodynamics & Navigation

Coordinate Geometry

Use distance, midpoint, slope, and transformations on the coordinate plane.

  • 30 min
  • Unlocks: Orbital Plotter
  • Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

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Key terms

midpoint
the point found by averaging corresponding coordinates

Step 2 of 7Try it firstDone

Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

Step 3 of 7LearnDone

The big idea

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The decision that matters

Distance uses differences; midpoint uses averages. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

Distance uses differences; midpoint uses averages. Keep exact values through the setup; round only when the stem asks.
Core relationship
d=(x2−x1)2+(y2−y1)2d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Use the formula only after matching every symbol to the diagram or equation.

Algebra or Desmos?

Use coordinate differences for distance and coordinate averages for midpoint. Desmos is helpful for checking a square root or graphing a point-slope equation through its required point.

Explore in Desmos

Check a distance and a perpendicular line

Use distance directly; for the line, graph point-slope form and check it passes through the point.

The calculator is live: change anything and see what happens.

Check a distance and a perpendicular line

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Trap

The main trap

Perpendicular slopes are negative reciprocals, not merely negatives.

Step 4 of 7Worked examplesDone

Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1

Problem

Find the distance from (−2,3)(-2,3) to (4,11)(4,11).
Coordinate changes are 6 and 8.

Which coordinate changes form the legs of the distance triangle?

The distance is 10.

Which calculation verifies it?

Worked example

Example 2: ACT twist

Problem

Find the equation through (2,−1)(2,-1) perpendicular to y=13x+5y=\frac13x+5.
Perpendicular slope is −3-3.

Which slope is perpendicular to 13\frac13?

The line is y=−3x+5y=-3x+5.

Why is its slope −3-3?

Faded example · you finish it

Example 3: You finish it

Problem

Find the midpoint of (−5,7)(-5,7) and (9,−3)(9,-3).
(−5+92,7+(−3)2)\left(\frac{-5+9}{2},\frac{7+(-3)}{2}\right)

Complete the calculation.

Desmos way

Verify the distance and point-slope equation

Use distance directly; for the line, graph point-slope form and check it passes through the point.

Verify the distance and point-slope equation

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Step 5 of 7PracticeDone

Practice

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Step 6 of 7Spot the mistakeDone

Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

Step 7 of 7Exit ticketDone

Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.