Question

find: \\(cos b\\)
the figure is not drawn to scale.
answer
\\(\frac{10}{24}\\) \\(\frac{24}{10}\\) \\(\frac{10}{26}\\) \\(\frac{24}{26}\\)

Explanation:

Step1: Recall cosine definition

In a right triangle, $\cos(\theta) = \frac{\text{adjacent side to } \theta}{\text{hypotenuse}}$.
For angle \( B \), identify adjacent and hypotenuse.
Adjacent side to \( B \): length \( 10 \)? Wait, no—wait, the right triangle has legs 10 and 24, hypotenuse 26 (since \( 10^2 + 24^2 = 100 + 576 = 676 = 26^2 \)).
Wait, angle \( B \): let's label the triangle. Right angle at the vertex with legs 10 (vertical) and 24 (horizontal), hypotenuse 26. So angle \( B \) is at the top, so adjacent side to \( B \) is the vertical leg (length 10? No, wait: adjacent side is the leg forming angle \( B \) along with hypotenuse. Wait, angle \( B \): the sides: opposite to \( B \) is 24, adjacent is 10, hypotenuse 26. Wait, no: in right triangle, angle \( B \): the two legs: one is opposite, one is adjacent. Wait, let's re-express:

Let the right triangle have vertices: right angle at \( C \), \( A \) at the bottom (with leg 24), \( B \) at the top (with leg 10), hypotenuse \( AB = 26 \). So angle at \( B \): the sides: adjacent to \( B \) is the leg \( BC = 10 \)? No, wait, adjacent side to angle \( B \) is the side that is part of angle \( B \) and not the hypotenuse. So angle \( B \) is between side \( BC \) (length 10) and hypotenuse \( AB \) (length 26), and the opposite side is \( AC = 24 \). Wait, no—cosine of angle \( B \) is adjacent over hypotenuse. Adjacent side to \( B \) is \( BC \) (length 10)? Wait, no, that can't be. Wait, maybe I mixed up. Wait, in a right triangle, for angle \( B \), the adjacent side is the leg that is connected to angle \( B \) and the right angle. Wait, the right angle is at \( C \), so \( AC = 24 \) (horizontal), \( BC = 10 \) (vertical), \( AB = 26 \) (hypotenuse). So angle at \( B \): the sides: \( BC = 10 \) (one leg), \( AB = 26 \) (hypotenuse), and \( AC = 24 \) (opposite leg). So adjacent side to \( B \) is \( BC = 10 \)? No, adjacent side is the leg that is adjacent (next to) angle \( B \), so between angle \( B \) and the right angle. Wait, angle \( B \) is at vertex \( B \), so the two sides forming angle \( B \) are \( BC \) (length 10) and \( AB \) (length 26). The other side is \( AC \) (length 24), opposite to \( B \). So cosine of \( B \) is adjacent ( \( BC = 10 \) ) over hypotenuse ( \( AB = 26 \) )? No, that would be \( \frac{10}{26} \), but that's not an option. Wait, maybe I labeled the triangle wrong. Wait, the legs are 10 and 24, hypotenuse 26. So angle \( B \): maybe the adjacent side is 24? Wait, no—wait, maybe the angle \( B \) has adjacent side 24? Wait, no, let's recalculate. Wait, the options are \( \frac{10}{24} \), \( \frac{24}{10} \), \( \frac{10}{26} \), \( \frac{24}{26} \). Wait, the correct adjacent side: let's re-express angle \( B \). Let's take angle \( B \): the sides: adjacent is the leg that is adjacent to \( B \), i.e., the leg that is not opposite. Wait, opposite to \( B \) is \( AC = 24 \), so adjacent is \( BC = 10 \)? No, that gives \( \frac{10}{26} \), but \( \frac{10}{26} = \frac{5}{13} \), not matching. Wait, maybe I mixed up angle \( A \) and \( B \). Wait, angle \( A \): adjacent side 24, hypotenuse 26, so \( \cos A = \frac{24}{26} \). But the question is \( \cos B \). Wait, angle \( B \): adjacent side is 10, hypotenuse 26? But \( \frac{10}{26} = \frac{5}{13} \), not an option. Wait, the options are \( \frac{10}{24} \), \( \frac{24}{10} \), \( \frac{10}{26} \), \( \frac{24}{26} \). Wait, maybe the triangle is labeled differently. Wait, maybe the right angle is at the vertex with leg 10 (horizontal) and 24 (vertical). So angle \( B \) is at the end of the 10 leg. So adjacent side to \( B \) is 24? No, that doesn't make sense. Wait, no—wait, the formula is \( \cos B = \frac{\text{adjacent to } B}{\text{hypotenuse}} \). Let's list the sides:

- Leg 1: 10 (let's say vertical)
- Leg 2: 24 (horizontal)
- Hypotenuse: 26 (since \( 10^2 + 24^2 = 26^2 \))

Angle \( B \) is at the top (end of vertical leg). So the sides around angle \( B \): vertical leg (length 10) and hypotenuse (length 26). The horizontal leg (length 24) is opposite to angle \( B \). Therefore, adjacent side to \( B \) is the vertical leg (length 10), hypotenuse is 26. So \( \cos B = \frac{10}{26} \)? But that's not simplifying to any of the options? Wait, no—wait, the options are \( \frac{10}{24} \), \( \frac{24}{10} \), \( \frac{10}{26} \), \( \frac{24}{26} \). Wait, maybe I made a mistake in identifying adjacent. Wait, maybe angle \( B \) is at the other vertex. Wait, maybe the right angle is at \( B \)? No, the diagram shows a right angle at the vertex with legs 10 and 24. Wait, the user's diagram: "the figure is a right triangle" with legs 10 (one leg), 24 (another leg), hypotenuse 26. Angle \( B \) is at the vertex with the leg of length 10. So adjacent side to \( B \): the leg of length 10? No, adjacent side is the side that is adjacent to angle \( B \), i.e., the side that is part of angle \( B \) and is not the hypotenuse. Wait, angle \( B \) is between the leg of length 10 and the hypotenuse. So adjacent side is 10, hypotenuse 26. So \( \cos B = \frac{10}{26} \)? But that's one of the options (third option: \( \frac{10}{26} \)). Wait, but let's check again. Wait, maybe I mixed up opposite and adjacent. Wait, in a right triangle, \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \), \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). So for angle \( B \), opposite side is 24, adjacent is 10, hypotenuse 26. So \( \cos B = \frac{10}{26} \), which simplifies to \( \frac{5}{13} \), but the option is \( \frac{10}{26} \) (third option). Wait, but let's check the options again. The options are:

1. \( \frac{10}{24} \)
2. \( \frac{24}{10} \)
3. \( \frac{10}{26} \)
4. \( \frac{24}{26} \)

So the correct one should be \( \frac{10}{26} \), which is the third option. Wait, but let's confirm the triangle labeling. If angle \( B \) is at the vertex where the leg of length 10 is, then adjacent side is 10, hypotenuse 26. So \( \cos B = \frac{10}{26} \).

Step2: Match with options

The third option is \( \frac{10}{26} \), so that's the answer.

Answer:

\(\frac{10}{26}\) (the third option, i.e., the option with \(\frac{10}{26}\))