SAT — Coordinate Plane Strategy

Chapter Overview & Weightage

Coordinate plane questions are guaranteed on every Digital SAT Math section. They test slopes, distance, midpoints, line equations, and basic conic sections (circles, parabolas).

Typical SAT weightage: $4-6$ questions per Math section.

Concept	Frequency
Linear equations and slopes	Every test
Distance/midpoint formulas	Every test
Circle equations	Most tests
Parabolas	Some tests
System of two lines	Every test

Key Concepts You Must Know

Slope formula: $m = (y_2 - y_1)/(x_2 - x_1)$
Slope-intercept form $y = mx + b$ vs point-slope form
Parallel lines have equal slopes; perpendicular lines have slopes whose product is $-1$
Distance formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
Midpoint formula: $M = ((x_1+x_2)/2, (y_1+y_2)/2)$
Circle equation: $(x-h)^2 + (y-k)^2 = r^2$
Parabola in vertex form: $y = a(x-h)^2 + k$

Important Formulas

Slope-intercept: $y = mx + b$

Point-slope: $y - y_1 = m(x - x_1)$

Standard: $Ax + By = C$ , slope $= -A/B$

$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

$(x-h)^2 + (y-k)^2 = r^2$

centre $(h, k)$ , radius $r$ . Complete the square if given general form.

Solved Previous Year Questions

PYQ 1 (Practice Test)

The line $y = 3x - 5$ is perpendicular to which line?

A line perpendicular to slope $3$ has slope $-1/3$ . Look for any line with slope $-1/3$ , e.g., $y = -x/3 + 2$ .

PYQ 2 (Practice Test)

A circle has equation $x^2 + y^2 - 6x + 4y - 12 = 0$ . Find the centre and radius.

Complete the square: $(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4 = 25$ .

So $(x-3)^2 + (y+2)^2 = 25$ . Centre $(3, -2)$ , radius $5$ .

PYQ 3 (Practice Test)

Find the distance between $(2, 3)$ and $(7, 15)$ .

$d = \sqrt{(7-2)^2 + (15-3)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ .

A 5-12-13 Pythagorean triple — SAT loves these.

Difficulty Distribution

Difficulty	% of SAT Qs	Typical type
Easy	$35\%$	Direct slope, distance plug-ins
Medium	$50\%$	Circle equations, parallel/perpendicular lines
Hard	$15\%$	System of equations with geometric interpretation

Expert Strategy

Memorise the Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. SAT distance questions often use these so the answer is a clean integer.

For circle questions, immediately recognise the standard form. If given general form, complete the square — never plug in points.

For “find the equation of a line through two points” questions, compute slope first, then plug one point into $y = mx + b$ .

Common Traps

Using the slope formula upside down: $(x_2 - x_1)/(y_2 - y_1)$ . Slope is rise over run, so $y$ on top.

Saying perpendicular slopes have product $1$ . They have product $-1$ . Negative reciprocals.

Forgetting to take the square root in distance formula. $d^2 = 25 + 144 = 169$ , but $d = 13$ , not $169$ .

Treating $(x-h)^2 + (y-k)^2 = r^2$ centre as $(-h, -k)$ . The centre is $(h, k)$ with the signs as written. So $(x-3)^2$ means $h = 3$ , not $-3$ .