AMC 10 Daily Practice Round 1

Complete problem set with solutions and individual problem pages

Problem 24 Medium

There is a right-angled triangular metal sheet \triangle ABC with legs AB = 4 cm and BC = 3 cm, where m\angle B = 90^\circ.

Now, a square is to be cut from the triangle as shown in the figure.

What is the side length of the square?

  • A.

    \frac{6}{7}

  • B.

    \frac{30}{37}

  • C.

    \frac{12}{7}

  • D.

    \frac{60}{37}

  • E.

    \frac{28}{37}

Answer:D

As shown in the figure, draw BP perpendicular to AC from point B, with P as the foot of the perpendicular.

 

Line BP intersects DE at Q.

∵{{A}_{\triangle ABC}}=\frac{1}{2}AB\cdot BC=\frac{1}{2}AC\cdot BP.

∴BP=\frac{AB\cdot BC}{AC}=\frac{4\times 3}{5}=\frac{12}{5}.

∵DE||AC,

∴m\angle BDE=m\angle A，m\angle BED=m\angle C,

∴\triangle BDE\backsim \triangle BAC,

∴\frac{DE}{AC}=\frac{BQ}{BP}.

Assume DE=x, \frac{x}{5}=\frac{\frac{12}{5}-x}{\frac{12}{5}},

x=\frac{60}{37},

Choose \boxed{\text{D}:\frac{60}{37}}.

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