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Technique: coordinate bash

Place the figure on coordinates and compute

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Technique: coordinate bashAMC 10 Step by Step

Name: Date: September 13, 2026

12 problems · 40 minutes · 6 / 1.5 / 0

1.2010 AMC 10A #7
Crystal has a running course marked out for her daily run. She begins this run by heading due north for one mile. She then runs northeast for one mile, then southeast for one mile. The last portion of her run takes her on a straight line back to where she started. How far, in miles, is this last portion of her run?
1. A)
  $1$
2. B)
  $\sqrt{2}$
3. C)
  $\sqrt{3}$
4. D)
  $2$
5. E)
  $2\sqrt{2}$
2.2011 AMC 10A #18
Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$ . What is the area inside Circle $C$ but outside circle $A$ and circle $B$ ?
1. A)
  $3 - \frac{\pi}{2}$
2. B)
  $\frac{\pi}{2}$
3. C)
  $2$
4. D)
  $\frac{3\pi}{4}$
5. E)
  $1+\frac{\pi}{2}$
3.2012 AMC 10B #14
Two equilateral triangles are contained in a square whose side length is $2\sqrt 3$ . The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus?
1. A)
  $\frac{3}{2}$
2. B)
  $\sqrt 3$
3. C)
  $2\sqrt 2 - 1$
4. D)
  $8\sqrt 3 - 12$
5. E)
  $\frac{4\sqrt 3}{3}$
4.2025 AMC 10B #13
The altitude to the hypotenuse of a $30{-}60{-}90^\circ$ right triangle is divided into two segments of lengths $x < y$ by the median to the shortest side of the triangle. What is the ratio $\tfrac{x}{x+y}$ ?
1. A)
  $\frac 37$
2. B)
  $\frac{\sqrt3}{4}$
3. C)
  $\frac 49$
4. D)
  $\frac{5}{11}$
5. E)
  $\frac{4\sqrt3}{15}$
5.2012 AMC 10B #23
A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?
1. A)
  $\frac{\sqrt{3}}{3}$
2. B)
  $\frac{2\sqrt{2}}{3}$
3. C)
  $1$
4. D)
  $\frac{2\sqrt{3}}{3}$
5. E)
  $\sqrt{2}$
6.2014 AMC 10A #23
A rectangular piece of paper whose length is $\sqrt3$ times the width has area $A$ . The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$ . What is the ratio $B:A$ ?
1. A)
  $1:2$
2. B)
  $3:5$
3. C)
  $2:3$
4. D)
  $3:4$
5. E)
  $4:5$
7.2015 AMC 10B #19
In $\triangle{ABC}$ , $\angle{C} = 90^{\circ}$ and $AB = 12$ . Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$ , and $W$ lie on a circle. What is the perimeter of the triangle?
1. A)
  $12+9\sqrt{3}$
2. B)
  $18+6\sqrt{3}$
3. C)
  $12+12\sqrt{2}$
4. D)
  $30$
5. E)
  $32$
8.2016 AMC 10A #21
Circles with centers $P, Q$ and $R$ , having radii $1, 2$ and $3$ , respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$ , respectively, with $Q'$ between $P'$ and $R'$ . The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$ ?
1. A)
  $0$
2. B)
  $\sqrt{6}/3$
3. C)
  $1$
4. D)
  $\sqrt{6}-\sqrt{2}$
5. E)
  $\sqrt{6}/2$
9.2017 AMC 10B #19
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$ . Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$ , and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$ . What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$ ?
1. A)
  $9:1$
2. B)
  $16:1$
3. C)
  $25:1$
4. D)
  $36:1$
5. E)
  $37:1$
10.2018 AMC 10B #24
Let $ABCDEF$ be a regular hexagon with side length $1$ . Denote by $X$ , $Y$ , and $Z$ the midpoints of sides $\overline {AB}$ , $\overline{CD}$ , and $\overline{EF}$ , respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle ACE$ and $\triangle XYZ$ ?
1. A)
  $\frac {3}{8}\sqrt{3}$
2. B)
  $\frac {7}{16}\sqrt{3}$
3. C)
  $\frac {15}{32}\sqrt{3}$
4. D)
  $\frac {1}{2}\sqrt{3}$
5. E)
  $\frac {9}{16}\sqrt{3}$
11.2018 AMC 10A #23
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths $3$ and $4$ units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is $2$ units. What fraction of the field is planted?
1. A)
  $\frac{25}{27}$
2. B)
  $\frac{26}{27}$
3. C)
  $\frac{73}{75}$
4. D)
  $\frac{145}{147}$
5. E)
  $\frac{74}{75}$
12.2024 AMC 10A #22
Let $\mathcal K$ be the kite formed by joining two right triangles with legs $1$ and $\sqrt3$ along a common hypotenuse. Eight copies of $\mathcal K$ are used to form the polygon shown below. What is the area of triangle $\Delta ABC$ ?
1. A)
  $2+3\sqrt3$
2. B)
  $\dfrac92\sqrt3$
3. C)
  $\dfrac{10+8\sqrt3}{3}$
4. D)
  $8$
5. E)
  $5\sqrt3$

Technique: coordinate bash — answersAMC 10 Step by Step

Name: Date: September 13, 2026

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Answers

#	Answer	Topic and the idea it turns on
1	C	Triangles: Area & Pythagorean — The northeast and southeast miles are perpendicular legs of a right isosceles triangle, so together they move her sqrt(2) east and 0 north; the return leg is sqrt(1 + 2).
2	C	Circles — Center C is sqrt(2) from A and B, so circle C overlaps each of them in a 90-degree lens of area pi/2 - 1.
3	D	Quadrilaterals & Polygon Areas — Each triangle has height 3, so the apexes are 6 - 2√3 apart; the side lines cross at height √3, making the other diagonal 2√3 - 2.
4	A	Triangle Centers & Cevians — The altitude foot and the intersection point both lie on one ray from the right-angle vertex, so the 4:3 split is read straight off their first coordinates.
5	D	Solid Geometry — The cut face is perpendicular to the space diagonal from the removed corner; that corner's height above the face is 1/√3, so √3 - 1/√3 remains.
6	C	Transformations & Symmetry — The fold line makes a 60-degree angle with the base, so corner B lands exactly on corner D and the overlap is an equilateral triangle of area A/3.
7	C	Circles — The center lies on the perpendicular bisectors of XY and ZW, i.e. of AB and AC, so it is the midpoint of AB; equal radii force AC = BC.
8	D	Circles — Tangent circles on a common line have horizontal center separations sqrt((r1+r2)^2 - (r1-r2)^2): here 2sqrt2 and 2sqrt6, so the centers are (-2sqrt2, 1), (0, 2), (2sqrt6, 3) and shoelace finishes.
9	E	Triangles: Area & Pythagorean — The big triangle is ABC plus three triangles with sides 4s and 3s at a 120-degree angle, each 12 times [ABC], so 1 + 36 = 37.
10	C	Quadrilaterals & Polygon Areas — Each side of XYZ cuts a corner off ACE through the midpoint of one side and the quarter point of another, removing 1/8 of ACE three times.
11	D	Triangles: Area & Pythagorean — Cut the field into the square, a base-5 height-2 triangle on the hypotenuse, and two right triangles along the legs; total area 6 forces s = 2/7.
12	B	Triangles: Area & Pythagorean — AB = 3 + 3 is the sum of two bases of isosceles triangles with legs √3 and apex 120°, and C lies 3√3/2 above AB, giving area 9√3/2.