AMC 10 · 2024 · #9
Grade 8 probabilityPick an answer.
Both regions live in the plane and the answer is a ratio of areas, so the picture is everything. Tool #1 (Draw a Diagram) reveals what each inequality really is: |x| + |y| ≤ 8 is a square (rotated 45°) with diagonals on the axes, and (x² + y² - 25)² ≤ 49 is an annulus centered at the origin. Tool #7 (Identify Subproblems) splits the calculation in three clean parts — area of B (rotated square), area of T (annulus), and the ratio. Tool #13 (Convert to Algebra) tidies the target inequality: let r² = x² + y², then (r² - 25)² ≤ 49 becomes |r² - 25| ≤ 7, i.e. 18 ≤ r² ≤ 32 — two concentric circles.
Find the area of B
B is a square with area 128.
An absolute-value sum is a tilted square — draw the four corner triangles and read off the area at a glance.
7.G.B.6Draw A DiagramRead T as an annulus
It says the squared radius lies between 18 and 32.
Squaring then bounding ≤ 49 is just |·| ≤ 7 — Grade 8 "square root and squaring" reverse each other.
Squaring a quantity and bounding the result is the same as bounding its distance from zero.
▸ Why?
A number and its opposite have the same square, so the square only ever reports the size.
▸ Why?
Points at a fixed distance from a centre form a circle, so a bound on that distance is a ring.
Area of the ring
Its area is fourteen pi.
Annulus = big circle minus small circle. The area uses r², so the bounds r² ∈ [18, 32] plug in directly.
7.G.B.4Identify SubproblemsCheck the ring fits inside
The outer radius just touches the square's side.
The picture shows the annulus snugly inside the diamond — the outer circle just touches the diamond's sides. Without this check, we couldn't use Area(T) as is.
8.G.B.7Draw A DiagramReduce and add
The probability is seven over 64 pi, so 71.
Geometric probability is just the area-ratio — reduce, then add the numerator and denominator.
6.RP.A.3Identify SubproblemsThis AMC 12 problem only needs Grade 8 square-root reasoning plus the Grade 7 circle-area formula that you already know!
More like this
Same archetype — closest grade level first.
