AMC 10 · 2004 · #12
Grade 8 geometry-2dPick an answer.
The two unknown points are pinned down by two facts each: they sit on y=x, and they are collinear with a known pair of points. Tool #4 turns both facts into equations — write A'=(a,a) and B'=(b,b) so that the line condition is already built in, leaving one unknown per point. Each line is easy to write down because A and B are on the y-axis, so the y-intercept is handed to us and only the slope has to be computed from C. Tool #1 keeps the configuration honest: after solving, the sketch has to be checked to confirm C really falls inside both segments and that A'B' lies along the 45° line.
Give the unknown points coordinates
A point on that line has matching coordinates, so each unknown costs one number.
Writing a point as (t,t) means it can never leave the line y=x, no matter what the algebra does.
6.EE.B.6Introduce A VariableFind the line through A and C
The first two known points give a line with intercept read straight off the axis: slope -1/2.
A point on the y-axis hands you the intercept for free, so one slope calculation finishes the line.
8.EE.B.6Introduce A VariableIntersect that line with y = x
Intersecting with the diagonal gives A' = (6,6).
Being on two different lines at once pins a point to their single crossing point.
8.EE.C.8Introduce A VariableRepeat for B and C
The same two moves on the other segment give B' = (4,4).
The steeper line from the higher point reaches y=x sooner, so B' lands closer to the origin than A'.
8.EE.C.8Introduce A VariableCheck C is inside both segments
Checking confirms the crossing really lies inside both segments.
Lines stretch forever but segments stop, so a crossing point still has to fall between the endpoints.
6.NS.C.7Draw A DiagramMeasure A'B'
Equal horizontal and vertical changes give length 2√(2), choice (B).
Both points sit on the 45° line, so their distance is always √2 times the horizontal gap between them.
Both points sit on the forty-five degree line, so their distance is root two times the horizontal gap.
▸ Why?
A 45-45-90 triangle has legs equal and its hypotenuse root two times as long, so the ratio never has to be measured.
▸ Why?
That ratio is fixed by the square on the hypotenuse equalling the two squares on the legs.
Write a point on the line y = x as (t, t), and each straight-line condition becomes one easy equation — here they give (6,6) and (4,4), a distance of 2 times the square root of 2.
- Give the unknown points coordinates
- Find the line through A and C
- Intersect that line with y = x
- Repeat for B and C
- Check C is inside both segments
- Measure A'B'