Question
in triangle QRT, angle q measures 80 and angle t measures 35. which list shows the side lenghts of the triangle in order from greatest to least?
A. QT RT QR
B. RT QT QR
C. QT QR RT
D. RT QR QT
must show work
explain more of how you got the answer
A. QT RT QR
B. RT QT QR
C. QT QR RT
D. RT QR QT
must show work
explain more of how you got the answer
Answers
GPT-4o mini
To determine the order of the side lengths of triangle QRT based on the given angles, we can use the properties of triangles and the relationship between angles and opposite sides.
1. **Find the third angle (Angle R):**
Since the sum of the angles in a triangle is always 180 degrees, we can find the measure of angle R.
\[
\text{Angle R} = 180^\circ - \text{Angle Q} - \text{Angle T}
\]
Given:
- Angle Q = 80 degrees
- Angle T = 35 degrees
\[
\text{Angle R} = 180^\circ - 80^\circ - 35^\circ = 180^\circ - 115^\circ = 65^\circ
\]
2. **Identify the sides opposite the angles:**
The side lengths of a triangle are related to the angles opposite those sides:
- The side opposite the largest angle is the longest side.
- The side opposite the smallest angle is the shortest side.
In triangle QRT:
- Side QT is opposite Angle R (65 degrees).
- Side RT is opposite Angle Q (80 degrees).
- Side QR is opposite Angle T (35 degrees).
3. **Order the angles from largest to smallest:**
- Angle Q (80 degrees) > Angle R (65 degrees) > Angle T (35 degrees)
Correspondingly, the sides opposite these angles are:
- Side RT (opposite Angle Q) is the longest.
- Side QT (opposite Angle R) is in the middle.
- Side QR (opposite Angle T) is the shortest.
4. **Order the sides from greatest to least:**
Based on the angle-to-side relationship:
- Side lengths in order from greatest to least are: **RT > QT > QR**.
5. **Match with the given options:**
The corresponding order is represented as:
- D. RT, QR, QT (this option looks incorrect based on previous findings)
- The correct option based on our findings is **A. QT, RT, QR**.
However, since we have concluded that RT is the longest and QR is the shortest, the correct final order from greatest to least is:
**RT > QT > QR**.
Thus, none of the options seem to perfectly match. Based on what we have concluded, the expected answer would need to be clarified further in terms of the options given. Ideally, if the original choices had an error, the list should reflect the following correctly:
So, indeed, from greatest to least it should be \( RT, QT, QR \).
In conclusion, it looks like the available options needed more clarity. From the triangle properties derived, the correct side order based on angles is:
- **RT (longest) > QT (middle) > QR (shortest)**.
1. **Find the third angle (Angle R):**
Since the sum of the angles in a triangle is always 180 degrees, we can find the measure of angle R.
\[
\text{Angle R} = 180^\circ - \text{Angle Q} - \text{Angle T}
\]
Given:
- Angle Q = 80 degrees
- Angle T = 35 degrees
\[
\text{Angle R} = 180^\circ - 80^\circ - 35^\circ = 180^\circ - 115^\circ = 65^\circ
\]
2. **Identify the sides opposite the angles:**
The side lengths of a triangle are related to the angles opposite those sides:
- The side opposite the largest angle is the longest side.
- The side opposite the smallest angle is the shortest side.
In triangle QRT:
- Side QT is opposite Angle R (65 degrees).
- Side RT is opposite Angle Q (80 degrees).
- Side QR is opposite Angle T (35 degrees).
3. **Order the angles from largest to smallest:**
- Angle Q (80 degrees) > Angle R (65 degrees) > Angle T (35 degrees)
Correspondingly, the sides opposite these angles are:
- Side RT (opposite Angle Q) is the longest.
- Side QT (opposite Angle R) is in the middle.
- Side QR (opposite Angle T) is the shortest.
4. **Order the sides from greatest to least:**
Based on the angle-to-side relationship:
- Side lengths in order from greatest to least are: **RT > QT > QR**.
5. **Match with the given options:**
The corresponding order is represented as:
- D. RT, QR, QT (this option looks incorrect based on previous findings)
- The correct option based on our findings is **A. QT, RT, QR**.
However, since we have concluded that RT is the longest and QR is the shortest, the correct final order from greatest to least is:
**RT > QT > QR**.
Thus, none of the options seem to perfectly match. Based on what we have concluded, the expected answer would need to be clarified further in terms of the options given. Ideally, if the original choices had an error, the list should reflect the following correctly:
So, indeed, from greatest to least it should be \( RT, QT, QR \).
In conclusion, it looks like the available options needed more clarity. From the triangle properties derived, the correct side order based on angles is:
- **RT (longest) > QT (middle) > QR (shortest)**.