AMC 8 Daily Practice - Triangle Properties

Complete problem set with solutions and individual problem pages

Problem 7 Easy

In rectangle ABCD, AB = 18cm and BC = 12cm. The two line segments AE and AF divide the area of the rectangle into three equal parts. What is the area of triangle AEF?

  • A.

    50cm^{2}

  • B.

    54cm^{2}

  • C.

    60cm^{2}

  • D.

    66cm^{2}

  • E.

    72cm^{2}

Answer:C

The area of rectangle ABCD is AB \times BC.

The area of \triangle ABE is \frac{1}{2} \times AB \times BE = \frac{1}{3} \times AB \times BC, so BE = \frac{2}{3}BC.

The area of \triangle ADF is \frac{1}{2} \times AD \times DF = \frac{1}{3} \times AB \times BC, so DF = \frac{2}{3}AB.

Connect EF.

​

 

The area of \triangle CEF is \frac{1}{2} \times CF \times CE = \frac{1}{2} \times \left(\frac{1}{3}AB\right) \times \left(\frac{1}{3}BC\right) = \frac{1}{18} \times \text{Area of rectangle } ABCD.

Therefore, the area of \triangle AEF is: \left(\frac{1}{3} - \frac{1}{18}\right) \times \text{Area of rectangle } ABCD = \frac{5}{18} \times AB \times BC

Substituting the values AB = 18cm and BC = 12cm: \text{Area of } \triangle AEF = \frac{5}{18} \times 18 \times 12 = 60cm^{2}.

Thus, the area of \triangle AEF is 60cm^{2}.

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