Question

in △abc and △def, ∠a = 60°, ∠d = 60°, ab = 6, ac = 9, and de = 3. find the missing side df.
a. 3.5 cm
b. 4.5 cm
c. 5.5 cm
d. 6 cm

Explanation:

Step1: Determine similarity

Since \(\angle A=\angle D = 60^{\circ}\), if \(\triangle ABC\sim\triangle DEF\) (by AA similarity if other angles are equal or by SAS similarity if sides are in proportion). Assume \(\triangle ABC\sim\triangle DEF\). Then \(\frac{AB}{DE}=\frac{AC}{DF}\).

Step2: Substitute values

Given \(AB = 6\), \(AC=9\), \(DE = 3\). Substitute into \(\frac{AB}{DE}=\frac{AC}{DF}\), we get \(\frac{6}{3}=\frac{9}{DF}\).

Step3: Solve for \(DF\)

Cross - multiply: \(6\times DF=3\times9\), so \(6DF = 27\), then \(DF=\frac{27}{6}=4.5\)

Answer:

b. \(4.5\mathrm{cm}\)