AMC 10 · 2016 · #13
Grade 10 geometry-2dPick an answer.
The plane is above the ground, so this is a three-dimensional picture and the two sightings live in two different vertical planes. The move that flattens it is dropping the plane straight down to a single ground point X: every angle of elevation then becomes a right triangle standing on the ground, and the two houses plus X form one ordinary triangle I can work with on paper. The compass words are the hidden gift here, because 'due north' and 'due west' force a right angle at X, which is exactly what makes the Pythagorean Theorem available. After that I name the altitude h, write both ground distances in terms of h, and the geometry collapses into one equation in one unknown.
Drop the plane onto the ground
Dropping it down makes a ground point.
The shadow point straight under the plane is the hinge that connects everything on the ground to the one height I want.
10.G-MG.A.1Visualize Spatial RelationshipsNorth and west meet at a right angle
The two directions meet at a right angle.
Compass words are geometry in disguise: north-south and east-west always cross at 90°.
10.G-CO.A.1Draw A DiagramTurn each elevation into a ground distance
Each elevation gives a ground distance.
A shallow 30° sighting means the viewer is far away, a steep 60° sighting means close, and the tangent ratio turns each angle into an exact multiple of the height.
The tangent of the sighting angle turns each elevation into a ground distance measured in units of the height.
▸ Why?
In a right triangle the tangent of an angle is the far leg divided by the near one.
▸ Why?
The two ground distances meet at a right angle, so their squares add to the square of the gap between the viewers.
One equation in one unknown
That leaves one equation in one unknown.
Once both legs are measured in units of h, the Pythagorean relation has only one thing left to solve for.
8.G.B.7Convert To AlgebraPin down √(30) between choices
Bounding the root gives 5.5, choice (D).
Squaring the candidates is easier than rooting the answer, and it settles which choice the irrational value hugs.
8.NS.A.2Eliminate PossibilitiesDrop the flying object straight down to the ground first: the shadow point turns a 3D sighting problem into flat right triangles, and compass words like north and west quietly hand you a right angle.
- Drop the plane onto the ground
- North and west meet at a right angle
- Turn each elevation into a ground distance
- One equation in one unknown
- Pin down √(30) between choices