Georgia Milestones Algebra: a complete guide to geometry and modeling connections

What this part of the course demands

This guide covers the Geometry and Modeling Connections, the A.GSR coordinate-geometry domain (about 25 percent of the EOC) together with the A.MM modeling process that threads through the whole course. The geometry here is approachable and reuses slope and the distance formula, so it is a reliable block of points, and the modeling process is the skill the constructed-response items reward across every domain. Each dot-point page carries its own worked Milestones-style questions: distance and midpoint, slope, parallel, and perpendicular lines, perimeter and area in the coordinate plane, and the mathematical modeling process.

Distance and midpoint

The distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ is the Pythagorean theorem in coordinates: the horizontal and vertical changes are the legs, the distance is the hypotenuse. The midpoint formula $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ averages the coordinates. Distance is one number (a length); the midpoint is a point.

Slope, parallel, and perpendicular lines

Slope measures steepness and rate of change. Parallel lines have equal slopes; perpendicular lines have negative reciprocal slopes (flip and negate, product $-1$). A parallel or perpendicular condition just fixes the slope, after which you write the line with point-slope form.

Perimeter and area in the coordinate plane

Perimeter is the sum of the side lengths (use the distance formula, or the coordinate difference for axis-parallel sides). Area uses the right formula: rectangle $\text{length} \times \text{width}$, triangle $\frac{1}{2} \times \text{base} \times \text{height}$. Perimeter is in linear units; area is in square units.

The mathematical modeling process

The five steps: define variables, build a model (linear for a fixed amount, exponential for a fixed percent, quadratic for a max/min), solve, interpret with units, and check reasonableness (whole counts, sensible signs, correct rounding). The model-family choice is the decision everything else depends on.

How this strand is examined

• Numeric entry. Compute a distance, midpoint, perpendicular slope, perimeter, or area.
• Multiple choice. Identify a parallel or perpendicular slope, or the model family for a situation.
• Hot spot. Plot a midpoint or identify a figure's features on the plane.
• Constructed response. Solve a coordinate-geometry or modeling problem with full work and units.

Check your knowledge

Work these as you would for credit on the EOC.

1. Find the distance between $(2, 3)$ and $(7, 15)$. (2 points)
2. Find the midpoint of the segment from $(-4, 6)$ to $(2, -2)$. (1 point)
3. What is the slope of a line perpendicular to $y = \frac{3}{4}x - 2$? (1 point)
4. Write the line through $(0, 5)$ parallel to $y = 3x - 1$. (1 point)
5. A rectangle has vertices $(1, 1), (6, 1), (6, 3), (1, 3)$. Find its perimeter and area. (2 points)
6. A right triangle has vertices $(0, 0), (10, 0), (0, 6)$. Find its area. (1 point)
7. A quantity grows 8 percent per year. Which model family fits? (1 point)