1.9 The Coordinate Plane; Graphs of Equations; Circles
Ideas in this lesson
- Use distance and Pythagorean reasoning
- Find midpoints and recover endpoints
- Read, plot, and check graphs of equations
- Find and interpret axis intercepts
- Test symmetry and transform coordinate figures
- Build circle equations from geometric information
- Complete squares and classify circle equations
- Describe and graph coordinate regions
- Write circle equations from diameter endpoints
- Use tangency to one or both coordinate axes
- Prove coordinate geometry and calculate areas
- Apply midpoints to triangles and quadrilaterals
Use distance and Pythagorean reasoning · Worked example
For \(A=(2,-2),B=(5,2),C=(5,-2)\), which reasoning establishes a right triangle?
Answer: Its squared side lengths are 9, 16, and 25, so the converse of the Pythagorean theorem applies.
Segment AC is horizontal with length 3; segment BC is vertical with length 4.
The remaining squared distance is \(3^2+4^2=25\), so AB has length 5.
Since \(9+16=25\), the angle at C is right.
The remaining squared distance is \(3^2+4^2=25\), so AB has length 5.
Since \(9+16=25\), the angle at C is right.