Question

bc =
round your answer to the nearest hundredth.
right triangle with right angle at c, side ac = 7, angle at a is 65 degrees, side bc is unknown (marked with ?)
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Explanation:

Step1: Identify triangle type and trigonometric ratio

We have a right - triangle \( \triangle ABC \) with \( \angle C = 90^{\circ} \), \( \angle A=65^{\circ} \) and \( AC = 7 \). We want to find the length of \( BC \). We can use the tangent function, which is defined as \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \) in a right - triangle. For \( \angle A \), the opposite side to \( \angle A \) is \( BC \) and the adjacent side is \( AC \). So \( \tan(\angle A)=\frac{BC}{AC} \).

Step2: Substitute values and solve for \( BC \)

We know that \( \angle A = 65^{\circ} \) and \( AC = 7 \). From the tangent formula \( \tan(65^{\circ})=\frac{BC}{7} \). To find \( BC \), we can multiply both sides of the equation by 7: \( BC = 7\times\tan(65^{\circ}) \).

We know that \( \tan(65^{\circ})\approx2.1445 \) (using a calculator). Then \( BC=7\times2.1445 = 15.0115\approx15.01 \) (rounded to the nearest hundredth).

Answer:

\( 15.01 \)