AMC 10 · 2015 · #20
Grade 11 geometry-2dPick an answer.
T' looks like it has two unknowns, a and b, but the fixed perimeter ties them together, so the base b alone decides the triangle. Writing the area in terms of b turns "same area" into a single equation in a single unknown. That equation is a cubic, which sounds bad until you notice that T itself is a member of the same family and therefore already solves it — dividing out the solution you were handed leaves a quadratic. A final size estimate of the square root decides which listed number is nearest.
Measure the triangle you were given
The given triangle's area and perimeter are easy.
The altitude of an isosceles triangle always hits the middle of the base, so the shape splits into two matching right triangles.
The altitude of an isosceles triangle always hits the middle of the base, splitting it into two matching right triangles.
▸ Why?
Two equal sides make the triangle a mirror image of itself about that line.
▸ Why?
A fold line meets what it folds at right angles and cuts it exactly in half.
Trade two unknowns for one
The perimeter trades two unknowns for one.
A fixed perimeter is one equation, and one equation costs you one unknown.
6.EE.B.6Introduce A VariableWrite the area using b only
The area is then a function of the base.
Squaring the leg and subtracting the half-base wipes out the squared terms, so what is left under the root is only linear.
9.A-SSE.A.2Convert To AlgebraTurn equal areas into a cubic
Equal areas give one cubic.
Squaring both sides is only a safe move when you already know both sides are positive, and here the range check guarantees it.
9.A-CED.A.1Convert To AlgebraThe given triangle is already a root
The given triangle factors straight out.
The triangle you were given is already an answer to your own equation, so you can divide it away and keep what is left.
11.A-APR.B.2Organize Information In More WaysSolve the leftover quadratic
The leftover quadratic gives the new base.
A quadratic offers two numbers, and only one of them can be the length of a side.
9.A-REI.B.4Convert To AlgebraCheck a real triangle exists
That triangle really exists.
Solving an equation only narrows the candidates; you still owe an actual triangle at the end.
7.G.A.2Eliminate PossibilitiesDecide between 3 and 4
Bounding the root gives 3, choice (A).
Trapping the square root between two whole numbers already puts b on the near side of the 3.5 boundary.
8.NS.A.2Extreme PrincipleFixing the perimeter leaves the base as the only free number, and the triangle you were handed already solves the resulting equation, so dividing it out hands you the other one.
- Measure the triangle you were given
- Trade two unknowns for one
- Write the area using b only
- Turn equal areas into a cubic
- The given triangle is already a root
- Solve the leftover quadratic
- Check a real triangle exists
- Decide between 3 and 4