12. Coordinate Geometry: Slope, Distance, and Intersections

Coordinate Geometry: Slope, Distance, and Intersections

Slope is change in \(y\) per change in \(x\):
\[ m=\frac{y_2-y_1}{x_2-x_1}. \]
Parallel nonvertical lines have equal slopes; perpendicular slopes are negative
reciprocals. In \(y=mx+b\), \(b\) is the y-intercept.

An intersection satisfies both line equations. Set their expressions for \(y\)
equal, solve for \(x\), substitute for \(y\), and verify the ordered pair.

Distance is the Pythagorean theorem on a coordinate grid:
\[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. \]
If endpoints define a circle's diameter, divide their distance by 2 for the
radius. An unknown coordinate may produce two symmetric values, so follow the
wording about positive values or all solutions.

Learning objectives

You will connect equation and graph features, solve line intersections, use
distance as a geometric tool, and recognize symmetric coordinate solutions.

Worked example

Find the intersection of \(y=2x+1\) and \(y=-x+10\). At the intersection, the
two y-values are equal:
\[ 2x+1=-x+10,\qquad 3x=9,\qquad x=3. \]
Then \(y=2(3)+1=7\). The point \((3,7)\) satisfies both equations.

Now find the distance from \((1,2)\) to \((7,10)\):
\[ d=\sqrt{(7-1)^2+(10-2)^2} =\sqrt{36+64}=10. \]
Sketching the horizontal change 6 and vertical change 8 reveals a 6-8-10 right
triangle and independently verifies the formula.

Common traps

Keep subtraction order consistent in both coordinate differences. Reversing
both is harmless after squaring; reversing only part of the formula or omitting
a square is not. An intersection answer must include both coordinates.

Mastery practice

Create two nonparallel lines with an integer intersection. Solve algebraically,
then substitute the point into both equations. Build a coordinate-distance
problem whose changes form a familiar Pythagorean triple.