AMC 10 · 2022 · #14
Grade 11 geometry-2dalgebraPick an answer.
There is no formula that reads off the angle between two slanted segments straight from an equation, so the work has to be broken into pieces that each have a formula. Tool #7 (Identify Subproblems) runs the whole solution, and it runs twice. First it says: before any angle can be discussed, the curve has to become three concrete points, so solve for the intercepts. Then it makes the move the problem is really built around — the angle at B is hard, but the y-axis already passes through B and slices that angle into two pieces, and each piece sits in a right triangle whose legs lie along the axes. Two easy angles instead of one hard one. Tool #1 (Draw a Diagram) is what makes that slicing visible and, just as importantly, shows in advance that the angle is sharp rather than wide, which later decides a sign. Tool #15 (Organize Information in More Ways) does the quiet conversion in the middle: the coordinates -5, 3 and -15 have to stop being positions and start being leg lengths 5, 3 and 15 before any trigonometry applies. Tool #4 (Introduce a Variable) finishes it. Neither half-angle is a nameable number of degrees, so give them letters, α and β, keep only their tangents, and let the tangent addition formula put them back together.
Turn the curve into points
Factoring gives the three points.
Factoring is the fastest route to where a parabola crosses the x-axis, because a product is zero exactly when one of its factors is.
9.A-SSE.B.3Identify SubproblemsPlot and size up the angle
Two points sit on opposite sides of the axis.
A sketch that is roughly to scale tells you ahead of time whether an angle is sharp or wide, which is a free check on every formula written afterwards.
6.NS.C.8Draw A DiagramCut the angle with the axis
The axis splits it into two right triangles.
An awkward angle turns into two friendly ones when a convenient line passes through its vertex — and here the y-axis was already there.
7.G.B.5Identify SubproblemsRead both tangents
Read a tangent off each triangle.
Coordinates are lengths in disguise: they are exactly the legs of the right triangle a point forms with the axes.
10.G-SRT.C.6Organize Information In More WaysGlue with the addition formula
The addition formula gives four sevenths.
The addition formula lets you combine two angles you can never name individually, as long as you know each one's tangent.
The addition formula combines two angles you can never name individually, as long as each one's tangent is known.
▸ Why?
An angle built from two others is measured by adding the pieces, so the combination is legitimate.
▸ Why?
Each tangent is the far leg over the near one, and coordinates hand both legs over directly.
When an angle in a coordinate picture is awkward, slide the nearest axis through its corner to cut it into two right-triangle angles, read off each tangent, and glue them back together with the tangent addition formula.
- Turn the curve into three points
- Plot the points and size up the angle
- Let the y-axis cut the angle in two
- Read each tangent off the legs
- Glue the angles back with tangent addition