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 triangle

The segment \(AC\) is perpendicular to the base \(BD\) from vertex \(A\), representing the altitude.

Identify the hypotenuse of the right triangle

In the right triangle \(\triangle ABC\) with the right angle at \(C\), the side opposite to the right angle is \(AB\).

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

The side adjacent to the angle \(\theta\) (which is \(\angle ABC\)) that is not the hypotenuse is \(BC\). However, looking at the options, we have \(BD\), \(AD\), \(AC\), and \(AB\). Let's re-examine the right triangle \(\triangle ABD\), which has a right angle at \(A\).
In the large right triangle \(\triangle ABD\) with the right angle at \(A\):

• The hypotenuse is \(BD\).
• The side adjacent to \(\theta\) is \(AB\).
• The side opposite to \(\theta\) is \(AD\).

Let's check the matching options:

• altitude: This is the perpendicular segment from a vertex to the opposite side, which is \(AC\).
• hypotenuse: In the large right triangle \(\triangle ABD\), the hypotenuse is \(BD\).
• adjacent: In \(\triangle ABD\), the side adjacent to \(\theta\) is \(AB\).
• opposite: In \(\triangle ABD\), the side opposite to \(\theta\) is \(AD\).

Verify the matching pairs

Let's verify this set of matches:

• altitude \(

ightarrow\) \(AC\)

• hypotenuse \(

ightarrow\) \(BD\)

• adjacent \(

ightarrow\) \(AB\)

• opposite \(

ightarrow\) \(AD\)

This matches all four available draggable blocks (\(BD\), \(AD\), \(AC\), \(AB\)) perfectly.

Answer:

The correct matching of each line segment with its name is:

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