AMC 8 · 2015 · #8
Grade 6 geometry-2dPick an answer.
AMC 8 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The hidden gate is figuring out which values of x are even allowed. Tool #1 (Draw a Diagram) shows it directly: lay the side of length 19 flat, then swing the side of length 5. The far tip can reach at most 19 + 5 = 24 (when both are stretched in the same direction) and at least 19 - 5 = 14 (when folded back), so x must satisfy 14 ＜ x ＜ 24 — those are the limits where the triangle flattens into a line. Tool #9 (Solve an Easier Related Problem) confirms the same rule with a friendly case (sides 3 and 4 allow 1 ＜ x ＜ 7). Once we know x ＜ 24, the perimeter P = 24 + x ＜ 48, and Tool #6 (Guess and Check) on the choices picks the smallest whole number that stays above every legal P.
Sketch it: lay the 19-side flat, attach the 5-side at one end, and swing it like a clock hand — the third side x closes the gap.
Drawing a picture turns the abstract "is this a triangle?" question into something you can see: when the 5 points straight out along the 19, the tip lands at 24; when it folds back along the 19, the tip lands at 14.
4.G.A.2Draw A DiagramAs the 5-side swings, x nears 24 or 14 but never reaches either — at those limits the three sides collapse into a straight line.
This is the Triangle Inequality discovered visually: the third side has to be longer than the difference of the other two and shorter than their sum.
4.G.A.2Draw A DiagramTest the rule on sides 3 and 4: the same swing gives 1 ＜ x ＜ 7, and x = 7 makes 3 + 4 = 7 collapse to a line — confirmed.
Testing the rule on small numbers — Tool #9's job — makes the inequality feel solid before we trust it on the real numbers.
4.G.A.2Solve An Easier Related ProblemThe perimeter is P = 24 + x, and since x ＜ 24 it stays strictly below 48 — reaching, say, 47.99 but never 48 itself.
Adding the same number to both sides of the inequality x ＜ 24 keeps it true — a Grade 6 inequality move.
6.EE.B.8Identify SubproblemsPick the smallest whole number above every perimeter: 47 fails (P could be 47.5), but 48 always beats P ＜ 48 — so the answer is 48 (D).
Checking the choices against the bound is the Tool #6 (Guess and Check) finish: the right answer is the smallest whole number sitting just above the open boundary.
6.EE.B.8Guess And CheckThis AMC 8 problem only needs Grade 6 inequality reasoning — once you see why the third side has to stay under 5 + 19 = 24, the perimeter cap of 48 falls right out!