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Coordinate Geometry

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

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Coordinate GeometryAMC 10 Step by Step

Name: Date: October 28, 2026

20 problems, easiest first

What you need to know

Distance, midpoint, slope. For $(x_1,y_1)$ and $(x_2,y_2)$ : distance $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ (the Pythagorean Theorem), midpoint $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ , slope $\frac{y_2-y_1}{x_2-x_1}$ . Perpendicular lines have slopes that multiply to $-1$ (when neither line is vertical). Example: from $(1,2)$ to $(4,6)$ , the distance is $\sqrt{9+16} = 5$ , the midpoint is $\left(\frac52, 4\right)$ , the slope is $\frac43$ , and a perpendicular line has slope $-\frac34$ .
Distance from a point to a line. From $(x_0,y_0)$ to the line $ax+by+c=0$ , it is $\dfrac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$ . A line is tangent to a circle exactly when this distance from the center equals the radius. Example: from $(0,0)$ to $3x+4y-10=0$ the distance is $\frac{10}{5} = 2$ , so the line is tangent to the circle of radius $2$ centered at $(0,0)$ .
Circles. The circle with center $(h,k)$ and radius $r$ is $(x-h)^2+(y-k)^2=r^2$ . Given $x^2+y^2+Dx+Ey+F=0$ instead, complete the square in $x$ and in $y$ to read off the center and radius. Example: $x^2+y^2-6x+4y-12=0$ becomes $(x-3)^2+(y+2)^2 = 25$ : center $(3,-2)$ , radius $5$ .
Reflections. Over the $x$ -axis, $(x,y)$ goes to $(x,-y)$ . Over $y=x$ , swap to get $(y,x)$ . Over $y=-x$ , swap and change both signs to get $(-y,-x)$ . Example: $(2,5)$ goes to $(2,-5)$ , $(5,2)$ , and $(-5,-2)$ .
Lattice points (points whose coordinates are both integers). If $a$ and $b$ are integers, the segment from $(0,0)$ to $(a,b)$ contains $\gcd(a,b)+1$ lattice points, counting both ends. Example: to $(6,9)$ , $\gcd = 3$ , so $4$ points: $(0,0)$ , $(2,3)$ , $(4,6)$ , $(6,9)$ .

How AMC 10 tests it

Early: the area of a polygon from its corners, where two lines cross, or the perimeter of a polygon with lattice-point corners.
"A line through $(a,b)$ cuts a rectangle (or two circles) into equal areas." The line passes through the center of symmetry.
Middle to late: a line tangent to a circle, or a circle through three points, using the distance formula and completing the square.

Standard approaches

Sketch, then write every point, line, and circle as coordinates or an equation before computing.
Symmetry first: for a centrally symmetric figure (one that looks the same after a half turn, like a rectangle), every line through its center cuts it into equal areas. So try the line through the given point and the center. For a convex figure like a rectangle, those are the only lines that do it.
For tangency, set the center-to-line distance equal to $r$ . It is faster than solving a quadratic.
For areas, use the Shoelace Theorem. For lattice points, use the 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. Step 1. Put the line in the form $ax+by+c=0$ . Multiply by $4$ : $4y = 3x + 4b$ , so $3x - 4y + 4b = 0$ .

Step 2. Tangency means the center $(2,1)$ is exactly $2$ from the line:
$\frac{|3\cdot 2 - 4\cdot 1 + 4b|}{\sqrt{3^2+4^2}} = 2, \quad\text{so}\quad \frac{|2+4b|}{5} = 2 \quad\text{and}\quad |2 + 4b| = 10.$

Step 3. Either $2+4b = 10$ , so $b = 2$ , or $2+4b = -10$ , so $b = -3$ . So there are two tangent lines.

Step 4. The sum is $2 + (-3) = -1$ .

The answer is $\boxed{\textbf{(B)}\ -1}$ .

Pitfalls

Sign mistakes when rewriting $y = mx + b$ as $ax + by + c = 0$ . Example: $y = 2x+1$ becomes $2x - y + 1 = 0$ , not $2x + y + 1 = 0$ .
Forgetting that $|\text{something}| = k$ has two solutions, $k$ and $-k$ . That is why a tangent problem usually has two lines.
Reflecting over $y = x$ by changing signs instead of swapping. $(2,5)$ goes to $(5,2)$ , not $(-2,-5)$ .

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)

1.2019 AMC 10B #5
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?
1. A)Triangle $A'B'C'$ lies in the first quadrant.
2. B)Triangles $ABC$ and $A'B'C'$ have the same area.
3. C)The slope of line $AA'$ is $-1$ .
4. D)The slopes of lines $AA'$ and $CC'$ are the same.
5. E)Lines $AB$ and $A'B'$ are perpendicular to each other.
2.2012 AMC 10B #3
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?
1. A) $(998,2012)$
2. B) $(1000,1988)$
3. C) $(1000,2024)$
4. D) $(1000,4012)$
5. E) $(1012,2012)$
3.2025 AMC 10B #6
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 $s+t$ ?
1. A) $18$
2. B) $19$
3. C) $20$
4. D) $21$
5. E) $22$
4.2025 AMC 10B #7
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 reach the box by walking toward an unlocked gate, passing through it, and walking toward the box. It happens that the total distance Frances would travel is the same via either unlocked gate. What is the value of $x$ ?
1. A) $3\frac27$
2. B) $3\frac37$
3. C) $3\frac47$
4. D) $3\frac57$
5. E) $3\frac67$
5.2021 AMC 10B #9
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 $b - a ?$
1. A) $1$
2. B) $3$
3. C) $5$
4. D) $7$
5. E) $9$
6.2019 AMC 10B #4
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?
1. A) $(-1,2)$
2. B) $(0,1)$
3. C) $(1,-2)$
4. D) $(1,0)$
5. E) $(1,2)$
7.2019 AMC 10A #7
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 $x+y=10 ?$
1. A) $4$
2. B) $4\sqrt{2}$
3. C) $6$
4. D) $8$
5. E) $6\sqrt{2}$
8.2017 AMC 10B #10
The lines with equations $ax-2y=c$ and $2x+by=-c$ are perpendicular and intersect at $(1, -5)$ . What is $c$ ?
1. A) $-13$
2. B) $-8$
3. C) $2$
4. D) $8$
5. E) $13$
9.2017 AMC 10B #8
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 $C$ ?
1. A) $(-8, 9)$
2. B) $(-4, 8)$
3. C) $(-4, 9)$
4. D) $(-2, 3)$
5. E) $(-1, 0)$
10.2016 AMC 10B #9
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 $BC$ ?
1. A) $4$
2. B) $6$
3. C) $8$
4. D) $10$
5. E) $16$
11.2015 AMC 10B #12
For how many integers $x$ is the point $(x,-x)$ inside or on the circle of radius $10$ centered at $(5,5)$ ?
1. A) $11$
2. B) $12$
3. C) $13$
4. D) $14$
5. E) $15$
12.2011 AMC 10A #9
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?
1. A) $ac+ad+bc+bd$
2. B) $ac-ad +bc-bd$
3. C) $ac+ad -bc-bd$
4. D) $-ac-ad +bc+bd$
5. E) $ac-ad -bc+bd$
13.2008 AMC 10B #14
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 $A$ ?
1. A) $\left( - \frac {10}{3}\sqrt {3},5\right)$
2. B) $\left( - \frac {5}{3}\sqrt {3},5\right)$
3. C) $\left(\sqrt {3},5\right)$
4. D) $\left(\frac {5}{3}\sqrt {3},5\right)$
5. E) $\left(\frac {10}{3}\sqrt {3},5\right)$
14.2006 AMC 10B #12
The lines $x=\frac{1}{4}y+a$ and $y=\frac{1}{4}x+b$ intersect at the point $(1,2)$ . What is $a+b$ ?
1. A) $0$
2. B) $\frac{3}{4}$
3. C) $1$
4. D) $2$
5. E) $\frac{9}{4}$
15.2006 AMC 10A #11
Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$ ?
1. A) $\text{the empty set}$
2. B) $\textrm{one point}$
3. C) $\textrm{two lines}$
4. D) $\textrm{a circle}$
5. E) $\textrm{the entire plane}$
16.2003 AMC 10B #11
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?
1. A) $2$
2. B) $5$
3. C) $7$
4. D) $12$
5. E) $20$
17.2024 AMC 10B #19
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"?
1. A) $4$
2. B) $5$
3. C) $6$
4. D) $7$
5. E) $9$
18.2023 AMC 10A #19
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 $|r-s|$ ?
1. A) $\frac{1}{4}$
2. B) $\frac{1}{2}$
3. C) $\frac{3}{4}$
4. D) $\frac{2}{3}$
5. E) $1$
19.2020 AMC 10B #13
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 $90^{\circ}$ left. Andy continues his progress, increasing his distance each time by $1$ unit and always turning left. What is the location of the point which Andy makes the $2020$ left turn?
1. A) $(-1030, -994)$
2. B) $(-1030, -990)$
3. C) $(-1026, -994)$
4. D) $(-1026, -990)$
5. E) $(-1022, -994)$
20.2017 AMC 10A #17
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 $\frac{PQ}{RS}$ ?
1. A) $3$
2. B) $5$
3. C) $3\sqrt{5}$
4. D) $7$
5. E) $5\sqrt{2}$

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.