AMC 10 · 2021 · #11

Grade 10 geometry-2d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
chord-perpendicular-from-centerpythagorean-theoremsystems-of-equationscircle-equation convert-to-algebraidentify-subproblems ↑ Prerequisites: pythagorean-theoremchord-perpendicular-from-center
📏 Medium solution 💡 3 insights
Problem
Two circles share the same centre, one of radius 17 and one of radius 19. A chord of the larger circle is placed so that exactly half of its length lies inside the smaller circle. Find the full length of that chord.

Pick an answer.

(A)
$12\sqrt{2}$
(B)
$10\sqrt{3}$
(C)
$\sqrt{17 \cdot 19}$
(D)
18
(E)
$8\sqrt{6}$
How to solve
Strategy Introduce a Variable

Nothing here is numbered yet, so name the two things that matter: the distance from the shared center to the chord, and half the chord's length. One perpendicular dropped from the center meets the chord at a point that is the midpoint of the whole chord and also the midpoint of the inside piece, which turns the words "half lies inside" into a second, shorter leg. Two right triangles then share that same distance leg, so two Pythagorean equations appear and the shared unknown can be cancelled instead of computed.

1STEP 1

Drop the perpendicular from the center

Drop the perpendicular from the centre.

OM ⊥ AB, AM = MB, PM = MQ
2STEP 2

Name the half-lengths

Both chords share one perpendicular.

AB = 2h, PQ = 1/2(2h) = h, PM = h/2, OM = d
3STEP 3

Write Pythagoras twice

Write Pythagoras twice.

d² + h² = 19² = 361, d² + (h/2)² = d² + h²/4 = 17² = 289
4STEP 4

Subtract to erase the distance

Subtracting erases the distance.

h² - h²/4 = 361 - 289 → 3/4h² = 72 → h² = 96
5STEP 5

Match the chord to a choice

The chord is eight root six.

(2h)² = 4h² = 384 = 64 · 6 = (8√(6))² → AB = 8√(6)
Answer
8√(6)
Back-substitute: d² = 361 - h² = 361 - 96 = 265, so d = √(265) ≈ 16.28. That is less than 17, which it must be, otherwise the chord would miss the smaller circle entirely and no part of it could be inside. Now measure both pieces: the chord is 2√(96) ≈ 19.60, and the inside piece is 2√(289 - 265) = 2√(24) ≈ 9.80, exactly half of 19.60, as required. The length is also comfortably below the larger circle's diameter of 38, as any chord must be. The four wrong choices all land between 16.97 and 18, the range you get from combining 17 and 19 without ever using the halving condition, so the exact system is what separates 8√(6) from them.
💡Key takeaway

Drop one perpendicular from the shared center: it halves the chord and the inside piece at the same point, so two right triangles share a leg and subtracting their equations gives the length.

  • Drop the perpendicular from the center
  • Name the half-lengths
  • Write Pythagoras twice
  • Subtract to erase the distance
  • Match the chord to a choice

More like this

Same archetype — closest grade level first.