Quant Lesson 8: Coordinate Geometry • GMATCourse.mba

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The Coordinate Plane

Every point on the coordinate plane is described by an ordered pair $(x, y)$. The x-axis is horizontal; the y-axis is vertical. The four quadrants are numbered I–IV counter-clockwise from the top-right.

Quadrant Signs

Q II

(−, +)

Q I

(+, +)

Q III

(−, −)

Q IV

(+, −)

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Lines, Slopes & Equations

Slope

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$

Slope-Intercept

$y = mx + b$

Point-Slope

$y-y_1 = m(x-x_1)$

Slope Relationships

Parallel lines: $m_1 = m_2$ (same slope)

Perpendicular: $m_1 \times m_2 = -1$ (negative reciprocal)

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Distance & Midpoint Formulas

Distance Formula

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

= Pythagorean theorem on the coordinate plane

Midpoint Formula

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

= Average of x-coords, average of y-coords

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Geometric Shapes on the Coordinate Plane

Key Coordinate Geometry Facts

▸Area of triangle with vertices: use the formula or base × height from coordinate differences.

▸A circle centered at $(h,k)$ with radius $r$: $(x-h)^2 + (y-k)^2 = r^2$.

▸Horizontal line: $y = c$ (slope = 0). Vertical line: $x = c$ (slope = undefined).

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10 Coordinate Geometry Traps

1. Slope sign and direction

Positive slope: rises left to right. Negative slope: falls left to right. Zero: horizontal. Undefined: vertical.

2. Perpendicular slope is NEGATIVE reciprocal

If $m=3$, perpendicular slope = $-1/3$, not just $1/3$.

3. Distance formula vs slope formula

Distance uses differences squared under a root. Slope is just the ratio of differences.

4. y-intercept in $y=mx+b$

$b$ is where the line crosses the y-axis (set $x=0$). Not where it crosses $x$-axis.

5. Finding x-intercept

Set $y=0$ in the equation to find where the line crosses the x-axis.

6. Horizontal/vertical slopes

Horizontal line: slope = 0 (NOT undefined). Vertical line: slope IS undefined.

7. Midpoint is NOT the intersection

Midpoint of segment AB is the average of the coordinates — it's NOT where the perpendicular bisector crosses AB.

8. Circle equation: sign of h, k

$(x-h)^2+(y-k)^2=r^2$: center is at $(h,k)$, not $(-h,-k)$.

9. Coordinate of intersection

To find where two lines meet, solve the system of their equations simultaneously.

10. Quadrant determination with inequalities

If $x<0$ and $y>0$, the point is in Quadrant II — be careful with negative multiplied by negative.

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10 GMAT Practice Questions

Q1 PS Difficulty: 550

What is the slope of the line passing through points $(2, 5)$ and $(6, 1)$?

(A) −1. $m = \frac{1-5}{6-2} = \frac{-4}{4} = -1$.

Q2 PS Difficulty: 550

A line has slope $\dfrac{2}{3}$ and passes through the point $(0, -4)$. What is its equation?

(B) The y-intercept is $b = -4$ (passes through $(0,-4)$). Equation: $y = \frac{2}{3}x - 4$.

Q3 PS Difficulty: 600

What is the distance between points $(-1, 3)$ and $(3, 0)$?

(C) 5. $d = \sqrt{(3-(-1))^2+(0-3)^2} = \sqrt{16+9} = \sqrt{25} = 5$.

Q4 PS Difficulty: 550

What is the midpoint of the segment connecting $(4, 7)$ and $(-2, 3)$?

(B) $(1, 5)$. $M = \left(\frac{4+(-2)}{2}, \frac{7+3}{2}\right) = (1, 5)$.

Q5 PS Difficulty: 600

Line $L$ has slope $-\dfrac{3}{4}$. What is the slope of any line perpendicular to $L$?

(C) $\frac{4}{3}$. Perpendicular slope = negative reciprocal of $-\frac{3}{4}$ = $\frac{4}{3}$.

Q6 PS Difficulty: 650

A circle is centered at the origin with radius 7. Which of the following points lies ON the circle?

(D) $(3, \sqrt{40})$. Check: $3^2 + (\sqrt{40})^2 = 9+40 = 49 = 7^2$. ✓

Q7 PS Difficulty: 600

What is the area of triangle with vertices at $(0,0)$, $(4,0)$, and $(0,3)$?

(A) 6. Base = 4 (along x-axis), height = 3 (along y-axis). Area = $\frac{1}{2}(4)(3) = 6$.

Q8 DS Difficulty: 650

Do lines $y = 2x + 3$ and $y = 2x - 5$ intersect?

(1) Both lines have the same slope.
(2) The y-intercepts are different.

(D) Each alone sufficient. (1): Same slope means parallel — parallel lines never intersect (unless identical). NO they don't intersect. Sufficient. (2): Different y-intercepts with same slope → parallel, distinct lines. No intersection. Sufficient. Both → (D).

Q9 PS Difficulty: 700

If the line $3x - 4y = 12$ crosses the x-axis at point $A$ and the y-axis at point $B$, what is the length of segment $AB$?

(C) 5. x-intercept (set y=0): $3x=12$ → $x=4$. So $A=(4,0)$. y-intercept (set x=0): $-4y=12$ → $y=-3$. So $B=(0,-3)$. Distance = $\sqrt{4^2+(-3)^2}=\sqrt{25}=5$.

Q10 PS Difficulty: 750

Line $m$ passes through $(-3, 0)$ and $(0, 4)$. Line $n$ is perpendicular to $m$ and passes through the origin. Where do lines $m$ and $n$ intersect?

(D). Slope of $m$: $\frac{4-0}{0-(-3)} = \frac{4}{3}$. Equation: $y = \frac{4}{3}x+4$. Perpendicular slope = $-\frac{3}{4}$. Line $n$ through origin: $y=-\frac{3}{4}x$. Set equal: $-\frac{3}{4}x = \frac{4}{3}x+4$. Multiply by 12: $-9x = 16x+48$ → $-25x=48$ → $x=-\frac{48}{25}$. $y=-\frac{3}{4}(-\frac{48}{25})=\frac{36}{25}$.

Lesson Summary — Key Takeaways

Slope = rise/run = Δy/Δx

Positive rises left-to-right. Negative falls. Zero = horizontal. Undefined = vertical.

Perpendicular: m₁ × m₂ = −1

If slope is 2/3, perpendicular slope is −3/2. Always negate AND flip.

Distance = Pythagorean theorem

$d = \sqrt{(\Delta x)^2+(\Delta y)^2}$. The coordinate plane is just a right triangle.

Circle: $(x−h)^2+(y−k)^2=r^2$

Center at $(h,k)$, radius $r$. Plug in any point to test if it lies on the circle.