AMC 10 · 2020 · #12

Grade 10 geometry-2d

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

coordinate-geometryslope-interceptrotation-isometryperpendicular-slopesvector-addition identify-subproblemsphysical-representationconvert-to-algebra ↑ Prerequisites: coordinate-geometryslope-intercept

📏 Long solution 💡 3 insights

Problem

In the coordinate plane, a line is pinned at the point (20, 20) and turned 45 degrees counterclockwise about that pin. Find the x-coordinate of the point where the turned line crosses the x-axis.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Draw a Diagram

Tool #7 (Subproblems): first test whether the pin (20, 20) sits on l. If it does, the rotated line still runs through it, and the whole problem shrinks to one question — what is the new slope? Tool #4 (Variable): turn the slope into a direction arrow u = (5, 3), because arrows are what rotations act on. Tool #17 (Spatial): a 90^° turn of an arrow is exact and easy — swap the coordinates, flip one sign. Tool #1 (Diagram): 45^° is half of 90^°, and the picture that halves an angle is a rhombus, whose diagonal is the sum of its two equal sides. Tool #13 (Algebra): convert the new direction to a slope and write k in point-slope form. Tool #3 (Eliminate): set y = 0 and match the result against the five choices.

1STEP 1

Check where the pin sits

The pin lies on the line itself.

3(20) - 5(20) + 40 = 60 - 100 + 40 = 0

2STEP 2

Reduce to one slope

Then only the slope is needed.

(20, 20) ∈ l → (20, 20) ∈ k

3STEP 3

Name the direction

Write the line's direction as an arrow.

3x - 5y + 40 = 0 → y = 3/5x + 8, u = (5, 3)

4STEP 4

Turn a quarter turn

A quarter turn just swaps coordinates.

(x, y) ↦ (-y, x), u = (5, 3) ↦ u^ ⊥ = (-3, 5)

5STEP 5

Halve the turn with a rhombus

Adding two equal arrows halves the angle.

|u| = |u^ ⊥ | = √(34) → u + u^ ⊥ = (5, 3) + (-3, 5) = (2, 8)

6STEP 6

Write the new line

Write the line with the new slope.

m_k = 8/2 = 4, y - 20 = 4(x - 20) → y = 4x - 60

7STEP 7

Cross the x-axis

Setting y to zero gives 15.

0 = 4x - 60 → 4x = 60 → x = 15 → (B)

Answer

Two independent checks agree. First, angles: line l has slope 3/5, an inclination of about 31^°; turning counterclockwise by 45^° should give about 76^°, and tan 76^° ≈ 4.0 — exactly the slope found, and it is correctly steeper than the original, as a counterclockwise turn from a shallow positive slope must be. Second, position: k runs through (20, 20), a point 20 units above the axis, with positive slope 4, so descending to the axis takes a run of 20/4 = 5 to the left, landing at 20 - 5 = 15. That the crossing must lie strictly left of x = 20 also rules out choices (C), (D) and (E) before any computation.

💡Key takeaway

When the pin sits on the line, rotating changes nothing but the slope. A 45^° turn is half of a quarter turn, so turn the direction arrow (5, 3) into (-3, 5) and add them — the rhombus diagonal (2, 8) lands exactly halfway, giving slope 4.

Check where the pin sits
Shrink the problem to one slope
Name the direction of line l
Turn the arrow a quarter turn
Halve the quarter turn with a rhombus
Read the new slope, write line k
Set y = 0 and match a choice