Topics / Geometry

Coordinate Geometry

Distance, slope, lines, circles in the plane, reflections over lines, lattice points

Coordinate GeometryAMC 10 Step by Step

Name: Date: September 13, 2026

Geometry · 41 problems have this as their main idea

primary-topic problems (3.2% of all)

What you need to know

Distance $\sqrt{(\Delta x)^2+(\Delta y)^2}$ , midpoint $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ , slope $\frac{\Delta y}{\Delta x}$ ; perpendicular slopes multiply to $-1$ .
Distance from $(x_0,y_0)$ to the line $ax+by+c=0$ is $\dfrac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$ . A line is tangent to a circle exactly when this distance equals the radius.
A circle is $(x-h)^2+(y-k)^2=r^2$ ; complete the square to read center and radius from $x^2+y^2+Dx+Ey+F=0$ .
Reflections: over the $x$ -axis $(x,y)\to(x,-y)$ ; over $y=x$ , $(x,y)\to(y,x)$ ; over $y=-x$ , $(x,y)\to(-y,-x)$ .
The segment from $(0,0)$ to $(a,b)$ with integer $a,b$ contains $\gcd(a,b)+1$ lattice points.

How AMC 10 tests it

Early: area of a polygon from its vertices, intersection of two lines, or the perimeter of a lattice polygon.
"A line through $(a,b)$ divides a rectangle (or two circles) into equal areas": the line passes through the center of symmetry.
Mid-to-late: a line tangent to a circle, or a circle through three points, via the distance formula and completing the square.

Standard approaches

Sketch, then write every object (point, line, circle) as an equation before computing.
Symmetry first: a line bisecting a centrally symmetric figure passes through its center.
For tangency, set the center-to-line distance equal to $r$ rather than solving a quadratic.
For areas use shoelace; for lattice points use $\gcd$ on segments and count row by row.

Worked example

The line $y = \frac34 x + b$ is tangent to the circle with center $(2, 1)$ and radius $2$ . What is the sum of all possible values of $b$ ?

(A) $-3$ (B) $-1$ (C) $0$ (D) $1$ (E) $2$

Solution. Write the line as $3x - 4y + 4b = 0$ . Tangency means the distance from $(2,1)$ to the line equals $2$ :
$\frac{|3\cdot 2 - 4\cdot 1 + 4b|}{\sqrt{3^2+4^2}} = 2 \implies |2 + 4b| = 10.$
So $4b = 8$ or $4b = -12$ , giving $b = 2$ or $b = -3$ . The sum is $2 + (-3) = -1$ . The answer is $\boxed{\textbf{(B)}\ -1}$ .

Pitfalls

Sign errors when rewriting $y = mx + b$ as $ax + by + c = 0$ for the distance formula.
Forgetting that $|\cdot| = k$ has two solutions, so a tangent problem usually has two lines.
Reflecting over $y = x$ by negating instead of swapping coordinates.

Traps that recur

Counting lattice points as the area n^2 instead of (n+1)^2, or double counting the shared y-axis column when finding the points of R union S. (2022 AMC 10A #25)
Taking an interval symmetric around 2/3, or forgetting that points on the line count, so 2/3 itself is the lower endpoint and 19/28 the upper. (2021 AMC 10B #25)
Using 20 as the radius because the problem says diameter 20, which replaces 10 by 20 in the distance equation and produces g = 80/3, a value that cannot be written in the required form. (2025 AMC 10A #20)
Summing the three segment lengths only once (8), or treating the region as a parallelogram spanned by just two of the segments. (2023 AMC 10B #24)

Problems, easiest first

2019 AMC 10B · #5Coordinate Geometry

Triangle $ABC$ lies in the first quadrant. Points $A$ , $B$ , and $C$ are reflected across the line $y=x$ to points $A'$ , $B'$ , and $C'$ , respectively. Assume that none of the vertices of the triangle lie on the line $y=x$ . Which of the following statements is not always true?

2012 AMC 10B · #3Coordinate Geometry

The point in the xy-plane with coordinates $(1000, 2012)$ is reflected across the line $y = 2000$ . What are the coordinates of the reflected point?

2025 AMC 10B · #6Coordinate Geometry

The line $y = \frac{1}{3}x + 1$ divides the square region defined by $0 \leq x \leq 2$ and $0 \leq y \leq 2$ into an upper and lower region. The line $x=a$ divides the lower region into two regions of equal area. Then $a$ can be written as $\sqrt{s} - t$ , where $s$ and $t$ are positive integers. What is $s+t$ ?

2025 AMC 10B · #7Coordinate Geometry

Frances stands $15$ meters directly south of a locked gate in a fence that runs east-west. Immediately behind the fence is a box of chocolates, located $x$ meters east of the locked gate. An unlocked gate lies $9$ meters east of the box, and another unlocked gate lies $8$ meters west of the locked gate. Frances can …

2021 AMC 10B · #9Coordinate Geometry

The point $P(a,b)$ in the $xy$ -plane is first rotated counterclockwise by $90^\circ$ around the point $(1,5)$ and then reflected about the line $y = -x$ . The image of $P$ after these two transformations is at $(-6,3)$ . What is $b - a ?$

2019 AMC 10A · #7Coordinate Geometry

Two lines with slopes $\dfrac{1}{2}$ and $2$ intersect at $(2,2)$ . What is the area of the triangle enclosed by these two lines and the line $x+y=10 ?$

2019 AMC 10B · #4Coordinate Geometry

All lines with equation $ax+by=c$ such that $a,b,c$ form an arithmetic progression pass through a common point. What are the coordinates of that point?

2017 AMC 10B · #10Coordinate Geometry

The lines with equations $ax-2y=c$ and $2x+by=-c$ are perpendicular and intersect at $(1, -5)$ . What is $c$ ?

2017 AMC 10B · #8Coordinate Geometry

Points $A(11, 9)$ and $B(2, -3)$ are vertices of $\triangle ABC$ with $AB=AC$ . The altitude from $A$ meets the opposite side at $D(-1, 3)$ . What are the coordinates of point $C$ ?

2016 AMC 10B · #9Coordinate Geometry

All three vertices of $\bigtriangleup ABC$ are lying on the parabola defined by $y=x^2$ , with $A$ at the origin and $\overline{BC}$ parallel to the $x$ -axis. The area of the triangle is $64$ . What is the length of $BC$ ?

2015 AMC 10B · #12Coordinate Geometry

For how many integers $x$ is the point $(x,-x)$ inside or on the circle of radius $10$ centered at $(5,5)$ ?

2011 AMC 10A · #9Coordinate Geometry

A rectangular region is bounded by the graphs of the equations $y=a, y=-b, x=-c,$ and $x=d$ , where $a,b,c,$ and $d$ are all positive numbers. Which of the following represents the area of this region?

2008 AMC 10B · #14Coordinate Geometry

Triangle $OAB$ has $O=(0,0)$ , $B=(5,0)$ , and $A$ in the first quadrant. In addition, $\angle ABO=90^\circ$ and $\angle AOB=30^\circ$ . Suppose that $OA$ is rotated $90^\circ$ counterclockwise about $O$ . What are the coordinates of the image of $A$ ?

2006 AMC 10A · #11Coordinate Geometry

Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$ ?

2006 AMC 10B · #12Coordinate Geometry

The lines $x=\frac{1}{4}y+a$ and $y=\frac{1}{4}x+b$ intersect at the point $(1,2)$ . What is $a+b$ ?

2003 AMC 10B · #11Coordinate Geometry

A line with slope $3$ intersects a line with slope $5$ at point $(10,15)$ . What is the distance between the $x$ -intercepts of these two lines?

2024 AMC 10B · #19Coordinate Geometry

In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the 12 entries will be "Possible"?

2023 AMC 10A · #19Coordinate Geometry

The line segment formed by $A(1, 2)$ and $B(3, 3)$ is rotated to the line segment formed by $A'(3, 1)$ and $B'(4, 3)$ about the point $P(r, s)$ . What is $|r-s|$ ?

2020 AMC 10B · #13Coordinate Geometry

Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$ -direction). Andy moves $1$ unit and then turns $90^{\circ}$ left. From there, Andy moves $2$ units (north) and then turns $90^{\circ}$ left. He then moves $3$ units (west) and again turns $90^{\circ}$ …

2017 AMC 10A · #17Coordinate Geometry

Distinct points $P$ , $Q$ , $R$ , $S$ lie on the circle $x^2+y^2=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\frac{PQ}{RS}$ ?

2017 AMC 10A · #12Coordinate Geometry

Let $S$ be a set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3,~x+2,$ and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description for $S?$

2016 AMC 10B · #14Coordinate Geometry

How many squares whose sides are parallel to the axis and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$ , the line $y=-0.1$ and the line $x=5.1?$

2015 AMC 10A · #17Coordinate Geometry

A line that passes through the origin intersects both the line $x=1$ and the line $y=1+\frac{\sqrt{3}}{3}x$ . The three lines create an equilateral triangle. What is the perimeter of the triangle?

2014 AMC 10A · #14Coordinate Geometry

The $y$ -intercepts, $P$ and $Q$ , of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\triangle APQ$ ?

2014 AMC 10A · #18Coordinate Geometry

A square in the coordinate plane has vertices whose $y$ -coordinates are $0$ , $1$ , $4$ , and $5$ . What is the area of the square?

2013 AMC 10A · #18Coordinate Geometry

Let points $A = (0, 0)$ , $B = (1, 2)$ , $C=(3, 3)$ , and $D = (4, 0)$ . Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$ . This line intersects $\overline{CD}$ at point $\bigg(\frac{p}{q}, \frac{r}{s}\bigg)$ , where these fractions are in lowest terms. What is $p+q+r+s$ ?

2013 AMC 10A · #16Coordinate Geometry

A triangle with vertices $(6, 5)$ , $(8, -3)$ , and $(9, 1)$ is reflected about the line $x=8$ to create a second triangle. What is the area of the union of the two triangles?

2012 AMC 10A · #21Coordinate Geometry

Let points $A = (0 ,0 ,0)$ , $B = (1, 0, 0)$ , $C = (0, 2, 0)$ , and $D = (0, 0, 3)$ . Points $E$ , $F$ , $G$ , and $H$ are midpoints of line segments $\overline{BD},\text{ } \overline{AB}, \text{ } \overline {AC},$ and $\overline{DC}$ respectively. What is the area of $EFGH$ ?

2009 AMC 10B · #17Coordinate Geometry

Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$ , divides the entire region into two regions of equal area. What is $c$ ?

2006 AMC 10B · #20Coordinate Geometry

In rectangle $ABCD$ , we have $A=(6,-22)$ , $B=(2006,178)$ , $D=(8,y)$ , for some integer $y$ . What is the area of rectangle $ABCD$ ?

2000 AMC 10 · #16Coordinate Geometry

The diagram shows $28$ lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$ . Find the length of segment $AE$ .

2025 AMC 10A · #20Coordinate Geometry

A silo (right circular cylinder) with diameter 20 meters stands in a field. MacDonald is located 20 meters west and 15 meters south of the center of the silo. McGregor is located 20 meters east and $g > 0$ meters south of the center of the silo. The line of sight between MacDonald and McGregor is tangent to the silo. …

2023 AMC 10B · #24Coordinate Geometry

What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w, v+4w)$ with $0\le u\le1$ , $0\le v\le1,$ and $0\le w\le1$ ?

2021 AMC Fall 10B · #17Coordinate Geometry

Distinct lines $\ell$ and $m$ lie in the $xy$ -plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\ell$ to point $P'$ , and then $P'$ is reflected about line $m$ to point $P''$ . The equation of line $\ell$ is $5x - y = 0$ , and the coordinates of $P''$ are $(4,1)$ . What is the equation of …

2021 AMC 10A · #19Coordinate Geometry

The area of the region bounded by the graph of $x^2+y^2 = 3|x-y| + 3|x+y|$ is $m+n\pi$ , where $m$ and $n$ are integers. What is $m + n$ ?

2021 AMC 10A · #24Coordinate Geometry

The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$ , where $a$ is a positive real number. What is the area of this region in terms of $a$ , valid for all $a > 0$ ?

2018 AMC 10A · #21Coordinate Geometry

Which of the following describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$ -plane intersect at exactly $3$ points?

2017 AMC 10B · #24Coordinate Geometry

The vertices of an equilateral triangle lie on the hyperbola $xy=1$ , and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?

2011 AMC 10B · #24Coordinate Geometry

A lattice point in an $xy$ -coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx +2$ passes through no lattice point with $0 < x \le 100$ for all $m$ such that $1/2 < m < a$ . What is the maximum possible value of $a$ ?

2022 AMC 10A · #25Coordinate Geometry

Let $R$ , $S$ , and $T$ be squares that have vertices at lattice points (i.e., points whose coordinates are both integers) in the coordinate plane, together with their interiors. The bottom edge of each square is on the $x$ -axis. The left edge of $R$ and the right edge of $S$ are on the $y$ -axis, and $R$ contains …

2021 AMC 10B · #25Coordinate Geometry

Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30$ , inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y = mx$ . The possible values of $m$ lie in an interval of length $\frac{a}{b}$ , where $a$ and $b$ are relatively …