THE COMPOUND // FIELD NOTES // WEEK 15
Circle angles, gas laws, and lead coefficients
Field Book
Lessons 56 to 59
Due
Test 13 due · Quiz 3 · Supplement 15
Method
Paper first, every step
Completing the square meets a lead coefficient: divide it out first and the machine runs as before.
What you have to be able to do
- Use crossing chords and arc angle relationships
- Apply the ideal gas law as a formula problem
- Complete the square when the lead coefficient is not one
The procedures
Angles in circlesField Book Lesson 56
- Two chords crossing inside a circle make vertical angles equal to half the sum of the two intercepted arcs.
- Draw the arcs first, then write the angle equation.
Ideal gas lawField Book Lesson 57
- PV = nRT is an abstract equation like any other.
- Isolate the asked variable before substituting numbers.
Lead coefficientsField Book Lesson 58
- If a is not 1, divide every term by a first.
- Then complete the square exactly as before.
Experimental dataField Book Lesson 59
- Draw the line through the data, then read slope and intercept from your line.
- Simultaneous equations with fractions or decimals: clear them first, then eliminate.
Worked: dividing out the lead
- Solve 2x2 + 8x + 6 = 0.
- Divide by 2: x2 + 4x + 3 = 0.
- Factor: (x + 3)(x + 1) = 0, so x = -3 or x = -1.
Terms on the record
| Term | Meaning |
|---|
| chord | a segment whose endpoints lie on a circle |
| lead coefficient | the coefficient on the squared term |
| best-fit line | the line drawn to follow the trend of plotted data |