AMC 10 · 2017 · #8

Grade 8 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

coordinate-geometryisosceles-triangleperpendicular-bisector symmetry-argumentspatial-visualization ↑ Prerequisites: coordinate-geometry

📏 Medium solution 💡 1 insight

Problem

In △ ABC the two sides from A are equal, AB=AC. The vertices A(11,9) and B(2,-3) are known, and the altitude dropped from A hits the opposite side BC at the point D(-1,3). Find the coordinates of the third vertex C.

Pick an answer.

(A)

(-8, 9)

(B)

(-4, 8)

(C)

(-4, 9)

(D)

(-2, 3)

(E)

(-1, 0)

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Tool #1 (Draw a Diagram): plotting A, B, and D turns the coordinate list into a shape whose symmetry I can see. Tool #17 (Visualize Spatial Relationships): an isosceles triangle folds onto itself across the altitude from its apex, which forces the foot D to be the midpoint of the base BC — that single fact is the whole problem. Tool #4 (Introduce a Variable): once D is the midpoint, I name C=(x,y) and use the midpoint relation to solve for it.

1STEP 1

Plot the known points

Plot A(11,9), B(2,-3), D(-1,3); since the altitude from A meets BC at D, points B, D, and C lie on one line with D between B and C.

A=(11,9), B=(2,-3), D=(-1,3)∈ BC

2STEP 2

Use the isosceles symmetry

Since AB=AC, folding along AD maps B onto C, so the altitude perpendicularly bisects the base and its foot D is the midpoint of BC.

AB=AC → D is the midpoint of BC

3STEP 3

Write the midpoint equations

Let C=(x,y). Since D is the midpoint of B and C, each coordinate of D averages those of B and C, giving one equation for x and one for y.

(2+x)/2=-1, (-3+y)/2=3

4STEP 4

Solve for C

Double D and subtract B: x=2(-1)-2=-4 and y=2(3)-(-3)=9, so C=(-4,9), choice (C).

x=2(-1)-2=-4, y=2(3)-(-3)=9 → C=(-4,9)

Answer

(-4, 9)

Check that D really is the midpoint of B(2,-3) and C(-4,9): the average is (

\frac{2+(-4)}{2}

\frac{-3+9}{2}

)=(-1,3)=D. The equal-sides condition also holds: AB=√(9²+12²)=15 and AC=√(15²+0²)=15, so AB=AC as required. Both checks confirm C=(-4,9), choice (C).

💡Key takeaway

In an isosceles triangle the altitude from the tip lands on the middle of the base, so D is the midpoint of BC and C sits just as far past D as B — giving C=(-4,9).

Plot the known points
Use the isosceles symmetry
Write the midpoint equations
Solve for C

Plot the known points

Use the isosceles symmetry

Write the midpoint equations

Solve for C

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