Question

21 in the diagram of right triangle ade below.
bc || de.
which ratio is always equivalent to the sine of ∠a?

1. ad/de
2. ae/ad
3. bc/ab
4. ab/ac

Explanation:

Step1: Recall the definition of sine in a right - triangle

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). In right - triangle \(ADE\), \(\sin\angle A=\frac{DE}{AD}\).
Since \(BC\parallel DE\), \(\triangle ABC\sim\triangle ADE\) (by the AA similarity criterion, as \(\angle A=\angle A\) (common angle) and \(\angle ACB=\angle AED = 90^{\circ}\) because \(BC\parallel DE\) and \(\angle AED = 90^{\circ}\)).

Step2: Use the property of similar triangles

For similar triangles \(\triangle ABC\) and \(\triangle ADE\), the ratios of corresponding sides are equal. That is, \(\frac{BC}{DE}=\frac{AB}{AD}=\frac{AC}{AE}\).
From the definition of sine in \(\triangle ABC\), \(\sin\angle A=\frac{BC}{AB}\) (in \(\triangle ABC\), \(\angle A\) has opposite side \(BC\) and hypotenuse \(AB\)) and in \(\triangle ADE\), \(\sin\angle A=\frac{DE}{AD}\).
Since \(\frac{BC}{AB}=\frac{DE}{AD}\) (from similarity).

Answer:

1. \(\frac{BC}{AB}\)