Question

use the position of \\(\theta\\) in \\(\delta abc\\) to match each line segment with its name.

bd ad ac ab

altitude
hypotenuse
adjacent
opposite

Explanation:

Identify the altitude of the overall triangle

Using the Triangle Altitude knowledge point, we identify the segment perpendicular from a vertex to the opposite side.
In \(\triangle ABD\), the segment \(AC\) is perpendicular to the base \(BD\) at point \(C\).
Thus, the altitude is \(AC\).

Identify the hypotenuse relative to \(\theta\)

Using the Right Triangle Trigonometry knowledge point, we analyze the right triangle \(\triangle ABC\).
The right angle is at vertex \(A\) for the large triangle \(\triangle ABD\), and vertex \(C\) is a right angle for \(\triangle ABC\).
In the right-angled triangle \(\triangle ABC\), the side opposite the right angle \(\angle ACB\) is \(AB\).
Thus, the hypotenuse of \(\triangle ABC\) is \(AB\).

Identify the adjacent side relative to \(\theta\)

Using the Right Triangle Trigonometry knowledge point, we find the side next to angle \(\theta\) in \(\triangle ABC\).
The side adjacent to \(\theta\) (which is \(\angle B\)) that is not the hypotenuse is \(BC\).
However, looking at the options: \(BD\), \(AD\), \(AC\), \(AB\).
Let's re-evaluate the context of \(\triangle ABC\) and the options.
The segment \(BC\) is not in the options.
Let's look at the larger right triangle \(\triangle ABD\) where the right angle is at \(A\).
In \(\triangle ABD\), the hypotenuse is \(BD\).
For angle \(\theta\) at vertex \(B\) in the right triangle \(\triangle ABD\):

• The hypotenuse is \(BD\).
• The opposite side is \(AD\).
• The adjacent side is \(AB\).
• The altitude of \(\triangle ABD\) relative to hypotenuse \(BD\) is \(AC\).

Match each segment with its name

Let's align the definitions for \(\triangle ABD\) with altitude \(AC\):

• altitude: \(AC\)
• hypotenuse: \(BD\)
• adjacent: \(AB\)
• opposite: \(AD\)

Answer:

• altitude: \(AC\)
• hypotenuse: \(BD\)
• adjacent: \(AB\)
• opposite: \(AD\)