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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)

1.2017 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) $5\sqrt{2}$
2.2016 AMC 10B #14
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 $x=5.1?$
1. A) $30$
2. B) $41$
3. C) $45$
4. D) $50$
5. E) $57$
3.2015 AMC 10A #17
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?
1. A) $2\sqrt{6}$
2. B) $2+2\sqrt{3}$
3. C) $6$
4. D) $3+2\sqrt{3}$
5. E) $6+\frac{\sqrt{3}}{3}$
4.2014 AMC 10A #14
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 $\triangle APQ$ ?
1. A) $45$
2. B) $48$
3. C) $54$
4. D) $60$
5. E) $72$
5.2014 AMC 10A #18
A square in the coordinate plane has vertices whose $y$ -coordinates are $0$ , $1$ , $4$ , and $5$ . What is the area of the square?
1. A) $16$
2. B) $17$
3. C) $25$
4. D) $26$
5. E) $27$
6.2013 AMC 10A #16
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?
1. A) $9$
2. B) $\frac{28}{3}$
3. C) $10$
4. D) $\frac{31}{3}$
5. E) $\frac{32}{3}$
7.2013 AMC 10A #18
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 $p+q+r+s$ ?
1. A) $54$
2. B) $58$
3. C) $62$
4. D) $70$
5. E) $75$
8.2012 AMC 10A #21
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 $EFGH$ ?
1. A) $\sqrt{2}$
2. B) $\frac{2\sqrt{5}}{3}$
3. C) $\frac{3\sqrt{5}}{4}$
4. D) $\sqrt{3}$
5. E) $\frac{2\sqrt{7}}{3}$
9.2009 AMC 10B #17
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 $c$ ?
1. A) $\frac 12$
2. B) $\frac 35$
3. C) $\frac 23$
4. D) $\frac 34$
5. E) $\frac 45$
10.2006 AMC 10B #20
In rectangle $ABCD$ , we have $A=(6,-22)$ , $B=(2006,178)$ , $D=(8,y)$ , for some integer $y$ . What is the area of rectangle $ABCD$ ?
1. A) $4000$
2. B) $4040$
3. C) $4400$
4. D) $40,000$
5. E) $40,400$
11.2000 AMC 10 #16
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 $AE$ .
1. A) $\frac{4\sqrt{5}}{3}$
2. B) $\frac{5\sqrt{5}}{3}$
3. C) $\frac{12\sqrt{5}}{7}$
4. D) $2\sqrt{5}$
5. E) $\frac{5\sqrt{65}}{9}$
12.2025 AMC 10A #20
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. The value of $g$ can be written as $\frac{a\sqrt{b}-c}{d}$ , where $a,b,c,$ and $d$ are positive integers, $b$ is not divisible by the square of any prime, and $d$ is relatively prime to the greatest common divisor of $a$ and $c$ . What is the value of $a+b+c+d$ ?
1. A) $119$
2. B) $120$
3. C) $121$
4. D) $122$
5. E) $123$
13.2023 AMC 10B #24
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 $0\le w\le1$ ?
1. A) $10\sqrt{3}$
2. B) $13$
3. C) $12$
4. D) $18$
5. E) $16$
14.2021 AMC 10A #19
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 $m + n$ ?
1. A) $18$
2. B) $27$
3. C) $36$
4. D) $45$
5. E) $54$
15.2021 AMC Fall 10B #17
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 line $m?$
1. A) $5x+2y=0$
2. B) $3x+2y=0$
3. C) $x-3y=0$
4. D) $2x-3y=0$
5. E) $5x-3y=0$
16.2021 AMC 10A #24
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 $a > 0$ ?
1. A) $\frac{8a^2}{(a+1)^2}$
2. B) $\frac{4a}{a+1}$
3. C) $\frac{8a}{a+1}$
4. D) $\frac{8a^2}{a^2+1}$
5. E) $\frac{8a}{a^2+1}$
17.2018 AMC 10A #21
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?
1. A) $a=\frac14$
2. B) $\frac14 < a < \frac12$
3. C) $a>\frac14$
4. D) $a=\frac12$
5. E) $a>\frac12$
18.2017 AMC 10B #24
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?
1. A) $48$
2. B) $60$
3. C) $108$
4. D) $120$
5. E) $169$
19.2011 AMC 10B #24
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 $a$ ?
1. A) $\frac{51}{101}$
2. B) $\frac{50}{99}$
3. C) $\frac{51}{100}$
4. D) $\frac{52}{101}$
5. E) $\frac{13}{25}$
20.2022 AMC 10A #25
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 $\frac{9}{4}$ as many lattice points as does $S$ . The top two vertices of $T$ are in $R \cup S$ , and $T$ contains $\frac{1}{4}$ of the lattice points contained in $R \cup S.$ See the figure (not drawn to scale). The fraction of lattice points in $S$ that are in $S \cap T$ is $27$ times the fraction of lattice points in $R$ that are in $R \cap T$ . What is the minimum possible value of the edge length of $R$ plus the edge length of $S$ plus the edge length of $T$ ?
1. A) $336$
2. B) $337$
3. C) $338$
4. D) $339$
5. E) $340$

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.