Question

which statements are true about triangle qrs? select three options.
the side opposite ∠q is ( overline{rs} ).
the side opposite ∠r is ( overline{rq} ).
the hypotenuse is ( overline{qr} ).
the side adjacent to ∠r is ( overline{sq} ).
the side adjacent to ∠q is ( overline{qs} ).
(image of right triangle qrs with right angle at s)

Explanation:

Step1: Analyze triangle QRS (right-angled at S)

In right - triangle \( \triangle QRS \) with \( \angle S = 90^{\circ} \):
- **Opposite side to an angle**: The side opposite an angle is the side that does not form the angle. For \( \angle Q \), the sides forming \( \angle Q \) are \( \overline{QS} \) and \( \overline{QR} \), so the side opposite \( \angle Q \) is \( \overline{RS} \).
- For \( \angle R \), the sides forming \( \angle R \) are \( \overline{RS} \) and \( \overline{QR} \), so the side opposite \( \angle R \) is \( \overline{QS} \), not \( \overline{RQ} \).
- **Hypotenuse**: The hypotenuse is the side opposite the right angle. The right angle is \( \angle S \), so the hypotenuse is \( \overline{QR} \).
- **Adjacent side to an angle**: The adjacent side to an angle (other than the right angle) is one of the sides that forms the angle (excluding the hypotenuse). For \( \angle R \), the sides forming \( \angle R \) are \( \overline{RS} \) and \( \overline{QR} \) (hypotenuse), so the adjacent side to \( \angle R \) is \( \overline{RS} \) (and also \( \overline{QR} \) is hypotenuse). Wait, the option says "The side adjacent to \( \angle R \) is \( \overline{SQ} \)". \( \overline{SQ} \) is opposite to \( \angle R \), so this is wrong. For \( \angle Q \), the sides forming \( \angle Q \) are \( \overline{QS} \) and \( \overline{QR} \) (hypotenuse), so the adjacent side to \( \angle Q \) is \( \overline{QS} \).

Step2: Evaluate each option

- Option 1: "The side opposite \( \angle Q \) is \( \overline{RS} \)" - Correct, as we analyzed.
- Option 2: "The side opposite \( \angle R \) is \( \overline{RQ} \)" - Incorrect, since side opposite \( \angle R \) is \( \overline{QS} \).
- Option 3: "The hypotenuse is \( \overline{QR} \)" - Correct, as hypotenuse is opposite right angle \( \angle S \).
- Option 4: "The side adjacent to \( \angle R \) is \( \overline{SQ} \)" - Incorrect, \( \overline{SQ} \) is opposite \( \angle R \).
- Option 5: "The side adjacent to \( \angle Q \) is \( \overline{QS} \)" - Correct, as \( \overline{QS} \) forms \( \angle Q \) (along with hypotenuse \( \overline{QR} \)).

So the correct statements are: "The side opposite \( \angle Q \) is \( \overline{RS} \)", "The hypotenuse is \( \overline{QR} \)", "The side adjacent to \( \angle Q \) is \( \overline{QS} \)". Wait, but the problem says "three options". Wait, maybe I made a mistake in adjacent side for \( \angle R \). Let's re - check.

Wait, in right - triangle notation, for \( \angle R \), the two non - hypotenuse sides are \( \overline{RS} \) (leg) and \( \overline{QS} \) (leg). The angle \( \angle R \) is between \( \overline{RS} \) and \( \overline{QR} \) (hypotenuse). So the adjacent side to \( \angle R \) can be \( \overline{RS} \), and the opposite side is \( \overline{QS} \). The option "The side adjacent to \( \angle R \) is \( \overline{SQ} \)" is wrong. For \( \angle R \), adjacent side is \( \overline{RS} \). But the option is about \( \overline{SQ} \).

Wait, maybe the triangle is labeled as \( Q \), \( R \), \( S \) with right angle at \( S \), so vertices are \( Q \), \( R \), \( S \), right - angled at \( S \). So:

- Opposite \( \angle Q \): \( RS \) (correct, option 1)
- Opposite \( \angle R \): \( QS \) (so option 2: "opposite \( \angle R \) is \( RQ \)" is wrong, \( RQ \) is hypotenuse)
- Hypotenuse: \( QR \) (correct, option 3)
- Adjacent to \( \angle R \): \( RS \) (option 4: "adjacent to \( \angle R \) is \( SQ \)" - \( SQ \) is opposite, so wrong)
- Adjacent to \( \angle Q \): \( QS \) (correct, option 5)

So the correct options are: "The side opposite \( \angle Q \) is \( \overline{RS} \)", "The hypotenuse is \( \overline{QR} \)", "The side adjacent to \( \angle Q \) is \( \overline{QS} \)". Wait, but the problem says "three options". Let's check again. Maybe the adjacent side to \( \angle R \) - the two sides forming \( \angle R \) are \( RS \) and \( RQ \) (hypotenuse), so adjacent sides are \( RS \) (and \( RQ \) is hypotenuse). The option says "adjacent to \( \angle R \) is \( SQ \)" - no. The adjacent to \( \angle Q \) is \( QS \) (correct). Opposite \( \angle Q \) is \( RS \) (correct). Hypotenuse is \( QR \) (correct). So these three options are correct.

Answer:

1. The side opposite \( \angle Q \) is \( \overline{RS} \)
2. The hypotenuse is \( \overline{QR} \)
3. The side adjacent to \( \angle Q \) is \( \overline{QS} \)