Difference between revisions of "2021 Fall AMC 10B Problems/Problem 17"
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The <math>45^{\circ}</math> angle is a little bit unwieldy in the coordinate plane, so we should try to make a <math>45-45-90</math> triangle. Let <math>A</math> be a point on <math>\ell</math>; to make <math>A</math> fit nicely in the diagram, let it be <math>(0.5,2.5)</math>. Now, let's draw a perpendicular to <math>\ell</math> through point <math>A</math>, intersecting <math>m</math> at point <math>B</math>. <math>OAB</math> is a <math>45-45-90</math> triangle, so <math>B</math> is a <math>90</math> degree counterclockwise rotation from <math>O</math> about <math>A</math>. Therefore, the coordinates of <math>B</math> are | The <math>45^{\circ}</math> angle is a little bit unwieldy in the coordinate plane, so we should try to make a <math>45-45-90</math> triangle. Let <math>A</math> be a point on <math>\ell</math>; to make <math>A</math> fit nicely in the diagram, let it be <math>(0.5,2.5)</math>. Now, let's draw a perpendicular to <math>\ell</math> through point <math>A</math>, intersecting <math>m</math> at point <math>B</math>. <math>OAB</math> is a <math>45-45-90</math> triangle, so <math>B</math> is a <math>90</math> degree counterclockwise rotation from <math>O</math> about <math>A</math>. Therefore, the coordinates of <math>B</math> are | ||
<cmath>(0.5+2.5,2.5-0.5) = (3,2).</cmath> | <cmath>(0.5+2.5,2.5-0.5) = (3,2).</cmath> | ||
− | So, <math>(2 | + | So, <math>(3,2)</math> is a point on line <math>m</math>, which we already know passes through the origin; therefore, <math>m</math>'s equation is <math>y=\frac{2x}{3} \implies \boxed{\textbf{(D) } 2x-3y = 0}.</math> |
~ihatemath123 | ~ihatemath123 |
Revision as of 22:05, 22 October 2022
Contents
Problem
Distinct lines and
lie in the
-plane. They intersect at the origin. Point
is reflected about line
to point
, and then
is reflected about line
to point
. The equation of line
is
, and the coordinates of
are
. What is the equation of line
Solution 1
Denote as the origin.
Even though the problem is phrased as a coordinate bash, that looks disgusting. Instead, let's try to phrase this problem in terms of Euclidean geometry, using the observation that , and that both
and
must pass through
in order to preserve the distance from
to the origin.
(
and
are just defined as points on lines
and
.)
Because of how reflections work, we have that
and
; adding these two equations together and using angle addition, we have that
. Since the sum of both sides combined must be
by angle addition,
This is helpful! We can now return to using coordinates, with this piece of information in mind:
The
angle is a little bit unwieldy in the coordinate plane, so we should try to make a
triangle. Let
be a point on
; to make
fit nicely in the diagram, let it be
. Now, let's draw a perpendicular to
through point
, intersecting
at point
.
is a
triangle, so
is a
degree counterclockwise rotation from
about
. Therefore, the coordinates of
are
So,
is a point on line
, which we already know passes through the origin; therefore,
's equation is
~ihatemath123
(We never actually had to use the information of the exact coordinates of ; as long as
, when we move
around, this will not affect
's equation.)
Solution 2
It is well known that the composition of 2 reflections , one after another, about two lines and
, respectively, that meet at an angle
is a rotation by
around the intersection of
and
.
Now, we note that is a 90 degree rotation clockwise of
about the origin, which is also where
and
intersect. So
is a 45 degree rotation of
about the origin clockwise.
To rotate 90 degrees clockwise, we build a square with adjacent vertices
and
. The other two vertices are at
and
. The center of the square is at
, which is the midpoint of
and
. The line
passes through the origin and the center of the square we built, namely at
and
. Thus the line is
. The answer is (D)
.
~hurdler, minor edits by nightshade2526
Solution 3
We know that the equation of line is
. This means that
is
reflected over the line
. This means that the line with
and
is perpendicular to
, so it has slope
. Then the equation of this perpendicular line is
, and plugging in
for
and
yields
.
The midpoint of and
lies at the intersection of
and
. Solving, we get the x-value of the intersection is
and the y-value is
. Let the x-value of
be
- then by the midpoint formula,
. We can find the y-value of
the same way, so
.
Now we have to reflect over
to get to
. The midpoint of
and
will lie on
, and this midpoint is, by the midpoint formula,
.
must satisfy this point, so
.
Now the equation of line is
~KingRavi
Solution 4
First, use Solution 1's method to get and that the angle between lines
and
is
. From here, note that the slope of line
is less than that of line
as otherwise
wouldn't even be close to
. Thus, line
is a
clockwise rotation of line
. Line
makes an angle of
with the positive x axis. Thus, line
makes an angle of
with the positive x axis. Thus, the slope of line
is
by the tangent addition formula. Since the slope of line
is
, its equation is
, which is choice
.
Video Solution
~hurdler
Video Solution 2 (by Interstigation)
https://www.youtube.com/watch?v=KdrYlPmqqv0
~Interstigation
See Also
2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.