Question

gonometric ratios given △abc ~ △xyz, what is the value of cos(z)? options: 5/12, 12/13, 5/13, 12/5 (or similar)

Explanation:

Step1: Identify corresponding angles in similar triangles

Since \(\triangle ABC \sim \triangle XYZ\), corresponding angles are equal. So, \(\angle Z=\angle C\). We need to find \(\cos(C)\) to get \(\cos(Z)\).

Step2: Recall the definition of cosine in a right triangle

In a right triangle, \(\cos(\theta)=\frac{\text{adjacent side to } \theta}{\text{hypotenuse}}\). For \(\triangle ABC\), right - angled at \(B\), the sides are: \(AB = 5\), \(BC=12\), and \(AC = 13\) (by Pythagorean theorem \(5^{2}+12^{2}=25 + 144=169 = 13^{2}\)).

For \(\angle C\), the adjacent side is \(BC = 12\) and the hypotenuse is \(AC=13\).

Step3: Calculate \(\cos(C)\)

Using the cosine formula, \(\cos(C)=\frac{BC}{AC}=\frac{12}{13}\). Since \(\angle Z=\angle C\), \(\cos(Z)=\cos(C)=\frac{12}{13}\).

Answer:

\(\frac{12}{13}\) (assuming one of the options is \(\frac{12}{13}\), if the options are \(\frac{5}{12}\), \(\frac{12}{13}\), \(\frac{5}{13}\), \(\frac{12}{5}\), then the correct option is the one with \(\frac{12}{13}\))