Cogito
Geometry · Chapter 7 · Lesson 1
Distance, Midpoint, and the Circle
Geometry done with coordinates.
10 problems · about 20 minutes · G-GPE.A.1, G-GPE.B.4, G-GPE.B.7
What this lesson teaches
The student uses the distance and midpoint formulas and writes the equation of a circle.
- Distance is Pythagoras applied to the coordinate gaps.
- The midpoint averages each coordinate.
- A circle equation states that every point is a fixed distance from the centre.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — Tangents and Chords: Radius 6, tangent 8. Distance to the centre?
Answer 10
Why 10.
Review — Inscribed Angles: Central 90°. Inscribed on the same arc?
Answer 45
Why 45°.
Warm Up
Straightforward practice. Get the method working first.
4 problemsFrom (2, 3) to (10, 9). Distance?
Answer 10
Why 10.
What is the distance formula really?
Answer The Pythagorean theorem in coordinates.
Why Pythagoras, written with coordinates.
From (0, 0) to (6, 8). Distance?
Answer 10
Why 36 + 64.
From (1, 1) to (4, 5). Distance?
Answer 5
Why 9 + 16.
Build It Up
The same ideas with more to keep track of.
2 problemsFrom (3, 7) to (9, 11). Midpoint x-coordinate?
Answer 6
Why Average 3 and 9.
x² + y² = 49. What is the radius?
Answer 7
Why r² = 49.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Distance: From (0, 0) to (3, 4). What is the distance?
Answer 5
Why 5.
The Midpoint: From (2, 4) to (8, 10). What is the x-coordinate of the midpoint?
Answer 5
Why 5.