In `/_\ A B C ,\ /_A=30^@,\ /_B=40^@a n d\ /_C=110^@`. The angles of the triangle formed by joining the mid-points of the sides of this triangle are

Options

A. (a)`70^@,\ 70^@,\ 40^@`
B. (b) `60^@,\ 40^@,\ 80^@`
C. (c) `30^@,\ 40^@,\ 110^@`
D. (d) `60^@,\ 70^@,\ 50^@`

Answer

To solve the problem, we need to find the angles of the triangle formed by joining the midpoints of the sides of triangle ABC, where the angles of triangle ABC are given as:

- Angle A = 30 degrees
- Angle B = 40 degrees
- Angle C = 110 degrees

### Step-by-Step Solution:

1. **Identify the Midpoints**:
Let P, Q, and R be the midpoints of sides BC, AC, and AB respectively.

2. **Apply the Midpoint Theorem**:
According to the midpoint theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. Therefore:
- QR is parallel to BC
- PQ is parallel to AB
- PR is parallel to AC

3. **Corresponding Angles**:
Since QR is parallel to BC, and AB acts as a transversal, we have:
- Angle ARQ = Angle ABC = 40 degrees (corresponding angles)
- Angle AQR = Angle ACB = 110 degrees (corresponding angles)

4. **Find Angle QRP**:
Using the property that the sum of angles on a straight line is 180 degrees, we have:
\[ \text{Angle ARQ} + \text{Angle QRP} + \text{Angle BRP} = 180 \]
Substituting the known values:
\[ 40 + \text{Angle QRP} + 30 = 180 \]
\[ \text{Angle QRP} = 180 - 70 = 110 \text{ degrees} \]

5. **Repeat for Other Angles**:
Now, consider PQ parallel to AB:
- Angle CQP = Angle CAB = 30 degrees (corresponding angles)
- Angle CPQ = Angle CBA = 40 degrees (corresponding angles)

Using the straight line property:
\[ \text{Angle RPB} + \text{Angle RPQ} + \text{Angle QPC} = 180 \]
Substituting the known values:
\[ 110 + \text{Angle RPQ} + 40 = 180 \]
\[ \text{Angle RPQ} = 180 - 150 = 30 \text{ degrees} \]

6. **Final Angles**:
For the last set of angles using PR parallel to AC:
- Angle BRP = Angle BAC = 30 degrees (corresponding angles)
- Angle BPR = Angle BCA = 110 degrees (corresponding angles)

Again using the straight line property:
\[ \text{Angle QR} + \text{Angle RQP} + \text{Angle CQP} = 180 \]
Substituting the known values:
\[ 110 + \text{Angle RQP} + 30 = 180 \]
\[ \text{Angle RQP} = 180 - 140 = 40 \text{ degrees} \]

7. **Conclusion**:
The angles of the triangle formed by the midpoints P, Q, and R are:
- Angle ARQ = 40 degrees
- Angle QRP = 110 degrees
- Angle RQP = 30 degrees

Thus, the angles of the triangle formed by joining the midpoints of the sides of triangle ABC are 30 degrees, 40 degrees, and 110 degrees.

### Final Answer:
The angles are **30 degrees, 40 degrees, and 110 degrees**.

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