2021 AMC Fall 10B Problem 17

P16-20~6 minPrint

Coordinate Geometry Transformations & Symmetry Set up the equation/formula and compute; no special trick needed Exploit symmetry to reduce work or pair up objects

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 $m?$

Show solution without answering

Key insight

Two flips over lines through the origin make a turn by twice their angle. P to P'' is a quarter turn clockwise, so m is l turned 45 degrees clockwise.

Solution

Two flips in a row over lines that cross at the origin act like one turn around the origin, by twice the angle between the lines. So find the turn that takes $P$ to $P''$ first.

1. What turn takes $P$ to $P''$ ? $P = (-1, 4)$ and $P'' = (4, 1)$ . Both are $\sqrt{1 + 16} = \sqrt{17}$ from the origin, which fits a turn. Test the quarter turn clockwise, which sends $(x, y)$ to $(y, -x)$ . It sends $(-1, 4)$ to $(4, 1)$ . Yes: it is a $90^\circ$ clockwise turn.
2. So the angle between the lines is half of that. Going from $\ell$ to $m$ is $45^\circ$ clockwise.
3. Turn line $\ell$ by $45^\circ$ clockwise. Line $\ell$ is $y = 5x$ , which goes through the point $(1, 5)$ . A $45^\circ$ turn is half a quarter turn. The quarter turn clockwise sends $(1, 5)$ to $(5, -1)$ . Halfway between the directions of $(1, 5)$ and $(5, -1)$ (which have the same length) is the direction of their sum:
$(1, 5) + (5, -1) = (6, 4),$
which points the same way as $(3, 2)$ .
4. Equation of $m$ . It goes through the origin and $(3, 2)$ , so its slope is $\frac23$ : $y = \frac23 x$ , or $2x - 3y = 0$ .

Check with the picture: $\ell$ is steep (slope $5$ ), and $m$ with slope $\frac23$ is lower, so $m$ really is turned clockwise from $\ell$ .

The answer is $\boxed{\textbf{(D)}\ 2x-3y=0}$ .

Why this works

A flip turns a shape over, and a second flip turns it back face up, so two flips over crossing lines make a turn around the crossing point. The turn angle is twice the angle between the lines. Spotting that $P$ to $P''$ is a simple quarter turn means you never have to find the middle point $P'$ at all.

Alternative approach

Find $P'$ directly and use it. Reflecting $P = (-1, 4)$ over $y = 5x$ gives $P' = \left(\frac{32}{13}, \frac{43}{13}\right)$ . (Check: the midpoint of $P$ and $P'$ is $\left(\frac{19}{26}, \frac{95}{26}\right)$ , which is on $y = 5x$ .) Line $m$ is the mirror between $P'$ and $P'' = (4, 1)$ , so it passes through their midpoint:
$\left(\frac{32/13 + 52/13}{2},\ \frac{43/13 + 13/13}{2}\right) = \left(\frac{42}{13}, \frac{28}{13}\right).$
It also passes through the origin, so its slope is $\frac{28}{42} = \frac23$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Reflecting in $\ell$ and then in $m$ , both through the origin, is the same as rotating about the origin by twice the angle from $\ell$ to $m$ .

Compare $P = (-1, 4)$ with $P'' = (4, 1)$ . Both are at distance $\sqrt{17}$ from the origin, and $(-1, 4) \mapsto (4, 1)$ is the map $(x, y) \mapsto (y, -x)$ , a $90^\circ$ clockwise rotation. So the angle from $\ell$ to $m$ is $45^\circ$ clockwise.

Line $\ell$ is $y = 5x$ , direction vector $(1, 5)$ . Rotating a vector $(a, b)$ clockwise by $45^\circ$ gives $\frac{1}{\sqrt2}(a + b,\ b - a)$ , so $(1, 5)$ becomes $\frac{1}{\sqrt2}(6, 4)$ , which points along $(3, 2)$ .

Thus $m$ passes through the origin with slope $\frac23$ : $y = \frac23 x$ , or $2x - 3y = 0$ .

The answer is $\boxed{\textbf{(D)}\ 2x-3y=0}$ .

Review in the AoPS books

Introduction to Geometry, Chapter 17: Analytic Geometry
Introduction to Geometry, Chapter 16: The More Things Change. . . (§16.1, §16.2, §16.3, §16.4)
Introduction to Algebra, Chapter 22: Special Manipulations (§22.3)
Introduction to Algebra, Chapter 8: Graphing Lines