Question

using the image provided, which of the following describes the side lengths of the triangle from least to greatest?
○ st < rs < rt
○ rt < ts < rs
○ sr < ts < rt
○ none of these

Explanation:

Step1: Find the third angle

In triangle \( RST \), the sum of angles in a triangle is \( 180^\circ \). Given \( \angle S = 70^\circ \), \( \angle R = 68^\circ \), so \( \angle T = 180 - 70 - 68 = 42^\circ \).

Step2: Relate angles to sides

In a triangle, the larger the angle, the longer the opposite side.
- Opposite \( \angle T = 42^\circ \) is \( RS \).
- Opposite \( \angle R = 68^\circ \) is \( ST \).
- Opposite \( \angle S = 70^\circ \) is \( RT \).

So the order of angles from least to greatest: \( \angle T (42^\circ) < \angle R (68^\circ) < \angle S (70^\circ) \). Thus, the order of sides (opposite these angles) from least to greatest: \( RS < ST < RT \)? Wait, no, wait. Wait, angle at T is 42, opposite RS; angle at R is 68, opposite ST; angle at S is 70, opposite RT. So angle order: \( 42^\circ (\angle T) < 68^\circ (\angle R) < 70^\circ (\angle S) \), so side order (opposite): \( RS < ST < RT \)? Wait, no, the options are about \( ST \), \( RS \), \( RT \). Wait the first option is \( ST < RS < RT \)? Wait no, let's re - check. Wait angle at T is 42, so side opposite (RS) is shortest. Angle at R is 68, side opposite (ST) is next. Angle at S is 70, side opposite (RT) is longest. So the order of sides from least to greatest: \( RS < ST < RT \)? Wait the options:

First option: \( ST < RS < RT \) – no.

Wait wait, maybe I mixed up. Let's label the triangle: vertices R, S, T. Angles: at S is 70, at R is 68, at T is 42. So:

- Side opposite \( \angle T \) (42°) is \( RS \).

- Side opposite \( \angle R \) (68°) is \( ST \).

- Side opposite \( \angle S \) (70°) is \( RT \).

So angle sizes: \( \angle T = 42^\circ < \angle R = 68^\circ < \angle S = 70^\circ \). Therefore, side lengths (opposite angles): \( RS (\text{opposite } \angle T) < ST (\text{opposite } \angle R) < RT (\text{opposite } \angle S) \). So the order of sides from least to greatest is \( RS < ST < RT \). Now let's check the options:

- Option 1: \( ST < RS < RT \) – incorrect.

- Option 2: \( RT < TS < RS \) – incorrect.

- Option 3: \( SR < TS < RT \) (SR is same as RS) – so \( RS < ST < RT \), which is correct? Wait the third option is \( SR < TS < RT \) (SR = RS, TS = ST). So \( RS < ST < RT \) is the correct order of sides from least to greatest. Wait but the first option selected was \( ST < RS < RT \), which is wrong. Wait maybe I made a mistake in angle - side correspondence.

Wait let's re - calculate the angle at T: sum of angles in triangle is 180. \( \angle S = 70 \), \( \angle R = 68 \), so \( \angle T=180 - 70 - 68 = 42 \). So:

- Angle at T: 42°, side opposite is \( RS \) (since in triangle RST, side opposite angle T is RS: because angle at T is between R and S, so side opposite is RS).

- Angle at R: 68°, side opposite is \( ST \) (angle at R is between S and T, so side opposite is ST).

- Angle at S: 70°, side opposite is \( RT \) (angle at S is between R and T, so side opposite is RT).

So the length of a side is proportional to the measure of its opposite angle. So:

Since \( \angle T = 42^\circ < \angle R = 68^\circ < \angle S = 70^\circ \), then the lengths of the sides opposite these angles are in the same order. So \( RS < ST < RT \) (because \( RS \) is opposite \( \angle T \), \( ST \) is opposite \( \angle R \), \( RT \) is opposite \( \angle S \)).

Now let's check the options:

1. \( ST < RS < RT \) – This would mean \( ST \) (opposite \( \angle R = 68^\circ \)) is less than \( RS \) (opposite \( \angle T = 42^\circ \)), which is false because \( 68^\circ>42^\circ \), so \( ST > RS \).

2. \( RT < TS < RS \) – \( RT \) is opposite \( \angle S = 70^\circ \), \( TS = ST \) is opposite \( \angle R = 68^\circ \), \( RS \) is opposite \( \angle T = 42^\circ \). So \( 70^\circ>68^\circ>42^\circ \), so \( RT > ST > RS \), so this order is false.

3. \( SR < TS < RT \) – \( SR = RS \) (opposite \( \angle T = 42^\circ \)), \( TS = ST \) (opposite \( \angle R = 68^\circ \)), \( RT \) (opposite \( \angle S = 70^\circ \)). Since \( 42^\circ<68^\circ<70^\circ \), then \( RS < ST < RT \), which is the same as \( SR < TS < RT \). So this option is correct. Wait but the first option was selected, but maybe that was a mistake. Wait the original selected option was \( ST < RS < RT \), but that's wrong. The correct order from least to greatest is \( RS (SR) < ST (TS) < RT \), so the third option \( SR < TS < RT \) is correct? Wait but let's check again.

Wait angle at T: 42, side RS (opposite) – length RS.

Angle at R: 68, side ST (opposite) – length ST.

Angle at S: 70, side RT (opposite) – length RT.

Since \( 42 < 68 < 70 \), then \( RS < ST < RT \), which is \( SR < TS < RT \) (since SR = RS, TS = ST). So the third option is correct. But the first option in the image was selected, which is wrong. Wait maybe the user's question is to find the correct option. Wait the options are:

1. \( ST < RS < RT \)

2. \( RT < TS < RS \)

3. \( SR < TS < RT \)

4. None of these

So based on angle - side relationship, the correct order of sides from least to greatest is \( SR (RS) < TS (ST) < RT \), so option 3 is correct.

Answer:

The correct option is C. \( SR < TS < RT \) (assuming the third option is labeled as C, but in the image, the third option is \( SR < TS < RT \))