AMC 10 Daily Practice Round 2

Complete problem set with solutions and individual problem pages

Problem 18 Medium

Refer to the figure below. In parallelogram ABCD, AD=8. Construct a circle from D such that it is tangent to both AB and BC with tangent points H and G, repsectively. The circle also intersects with AD and DC at E and F, respectively. If 5AE=4DE and 8CF=DF, determine the perimeter of parallelogram ABCD.

  • A.

    32

  • B.

    34

  • C.

    36

  • D.

    38

  • E.

    40

Answer:C

Let AB=x, then CF=\frac{x}{9}. Also, AE=\frac{4}{9}\cdot 8=\frac{32}{9} and ED=\frac{5}{9}\cdot 8=\frac{40}{9}.

∵ C{{G}^{2}}=CF\cdot CD=\frac{{{x}^{2}}}{9},

∴ CG=\frac{x}{3}.

∵ A{{H}^{2}}=AE\cdot AD=\frac{32}{9}\cdot 8,

∴ AH=\frac{16}{3} and HB=x-\frac{16}{3}.

By BC=BG+GC=BH+GC, we have 8=x-\frac{16}{3}+\frac{x}{3},

∴ x=10,

∴ P_{ABCD}=\left( 8+10 \right)\cdot 2=36.

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