Question

given right triangle abc, what is the value of tan(a)?
options:
$\frac{5}{13}$
$\frac{12}{13}$
$\frac{12}{5}$
$\frac{13}{12}$
(image of right triangle abc with right angle at c, ac = 10, bc = 24, ab = 26)

Explanation:

Step1: Recall tangent definition

In a right triangle, \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\) to \(\theta\).

Step2: Identify sides for \(\angle A\)

For \(\angle A\), opposite side is \(BC = 24\), adjacent side is \(AC = 10\)? Wait, no, wait. Wait, right triangle \(ABC\) with right angle at \(C\). So \(\angle C = 90^\circ\). So for \(\angle A\), the opposite side is \(BC\) and adjacent is \(AC\)? Wait, no, let's check the lengths. Wait, \(AC = 10\), \(BC = 24\), \(AB = 26\) (since \(10^2 + 24^2 = 100 + 576 = 676 = 26^2\), so it's a right triangle at \(C\)). So \(\angle A\): the opposite side to \(\angle A\) is \(BC = 24\)? Wait, no, \(\angle A\) is at vertex \(A\), so the sides: \(AC\) is adjacent (since it's one leg forming \(\angle A\)), \(BC\) is the other leg (opposite to \(\angle A\)), and \(AB\) is hypotenuse. Wait, no, wait: in triangle \(ABC\), right-angled at \(C\), so:

• \(\angle A\) is at \(A\), so the sides:
• Adjacent to \(\angle A\): \(AC\) (length 10)
• Opposite to \(\angle A\): \(BC\) (length 24)

Wait, but then \(\tan(A)=\frac{\text{opposite}}{\text{adjacent}}=\frac{BC}{AC}=\frac{24}{10}=\frac{12}{5}\). Ah, that's one of the options. So let's confirm:

In a right triangle, tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. For \(\angle A\) in right triangle \(ABC\) (right-angled at \(C\)):

• Opposite side to \(\angle A\): \(BC = 24\)
• Adjacent side to \(\angle A\): \(AC = 10\)

So \(\tan(A)=\frac{BC}{AC}=\frac{24}{10}=\frac{12}{5}\).

Answer:

\(\frac{12}{5}\) (the option with \(\frac{12}{5}\))