AMC 10 · 2003 · #22
Grade 10 geometry-2d
Pick an answer.
Chasing PQ directly means tracking two moving feet at once, which is messy. The move that collapses the problem is to stop looking at PQ and look at something equal to it (Tool #16): because the diagonals are perpendicular, O, P, N, Q form a rectangle, and PQ is a diagonal of that rectangle, so PQ = ON. Setting the centre at the origin with the diagonals as axes makes this visible in one line (Tools #1 and #4). Minimising ON is then the standard shortest-distance-to-a-line question, whose answer is the perpendicular from O to AB (Tool #14). That perpendicular length comes out fastest by computing the area of triangle AOB two different ways (Tool #15). The final step is arithmetic comparison against the five choices (Tool #3), since the question asks only which is nearest.
Put the centre at the origin
The diagonals bisect each other at right angles, so put the centre at the origin with halves 8 and 15.
A rhombus is built out of four copies of one right triangle, so laying its diagonals on the axes puts every useful right angle where it can be read off.
10.G-CO.C.11Draw A DiagramFind the side length
The side is the hypotenuse of an 8-15 right triangle, so it is 17.
Half of each diagonal makes the two legs, so the side of the rhombus is just the hypotenuse they determine.
8.G.B.7Introduce A VariablePQ equals ON
The centre and the two feet form a rectangle, so the segment always equals the distance from the centre.
The two perpendicular feet are just the shadows of N on the two axes, and the distance between the shadows is the same as the distance from N to the corner.
10.G-CO.C.9Change Focus Count The ComplementMinimise the distance from O to side AB
So minimise the distance to the side; the perpendicular foot really does land inside the segment.
A perpendicular is the shortest path to a line, and here the foot lands inside the side rather than off its end, so nothing blocks it.
10.G-SRT.B.5Extreme PrincipleGet the altitude by counting area twice
Counting the triangle's area two ways gives the altitude 120/17.
One triangle has one area, so pairing a different base with its own height gives a free equation for the height you want.
One triangle has one area, so pairing a different base with its own height gives a free equation for that height.
▸ Why?
A triangle covers half of base times height whichever side you call the base.
▸ Why?
Two descriptions of the very same area must agree, so setting them equal is a true equation.
Compare with the five choices
That is about 7.06, nearest to 7, choice (C).
"Closest to" is settled by comparing distances, not by rounding habits, so compute every gap and pick the smallest.
6.NS.C.7Eliminate PossibilitiesWhen two perpendiculars box a point into a rectangle, the segment joining their feet is the same length as the segment back to the corner — so a hard moving distance becomes an easy one.
- Put the centre at the origin
- Find the side length
- PQ equals ON
- Minimise the distance from O to side AB
- Get the altitude by counting area twice
- Compare with the five choices
More like this
Same archetype — closest grade level first.
