Question

Six points $A,$ $B,$ $C,$ $D,$ $E,$ and $F$ lie in a straight line in that order. Suppose that $G$ is a point not on the line and that $AC = 26,$ $BD = 22,$ $CE = 31,$ $DF = 33,$ $AF = 73,$ $CG = 40,$ and $DG = 30.$ Find the area of $\triangle BGE.$

Answer

Answer: 468

Concepts:coordinate geometry distance formula triangle area

Difficulty rating: 2010

Solution:

Place the line on a number line with $A = 0.$ Then $C = 26,$ $E = 26 + 31 = 57,$ $F = 73,$ $D = 73 - 33 = 40,$ and $B = 40 - 22 = 18.$

Write $G = (x, y).$ From $CG = 40$ and $DG = 30,$ $(x - 26)^2 + y^2 = 1600,$ $(x - 40)^2 + y^2 = 900.$ Subtracting gives $14(2x - 66) = 700,$ so $x = 58,$ and then $y^2 = 1600 - 32^2 = 576,$ so $G$ is at height $24$ above the line.

Since $B$ and $E$ both lie on the line, $BE = 57 - 18 = 39$ is a base with height $24,$ so the area is $\frac{1}{2} \cdot 39 \cdot 24 = 468.$

2025 AIME II Problem 1

Answer: 468

Difficulty rating: 2010

Solution:

Problem 1 in Other Years