To solve the question step by step, we will analyze each statement (P, Q, R, S, T) and determine if they are true or false.
### Step 1: Analyze Statement P
**Statement P**: In a triangle ABC in which AB = AC, the altitude AD bisects BC.
1. **Draw Triangle ABC**: Start by drawing triangle ABC with AB = AC, making it an isosceles triangle.
2. **Draw Altitude AD**: Draw the altitude from A to BC, which meets BC at point D.
3. **Congruent Triangles**: By the properties of isosceles triangles, triangles ADB and ADC are congruent (using the SAS criterion: AB = AC, AD = AD, and angle ADB = angle ADC).
4. **Conclusion**: Since triangles ADB and ADC are congruent, it follows that BD = DC, which means AD bisects BC.
**Result**: True
### Step 2: Analyze Statement Q
**Statement Q**: The sum of any two sides of a triangle is greater than twice the median drawn to the third side.
1. **Draw Triangle ABC**: Draw triangle ABC.
2. **Draw Median AD**: Draw the median from A to BC, meeting BC at point D.
3. **Construct Point E**: Extend AD to point E such that D is the midpoint of AE.
4. **Use Triangle Inequality**: According to the triangle inequality, AB + AC > AD + AD, which simplifies to AB + AC > 2AD.
5. **Conclusion**: This statement is true.
**Result**: True
### Step 3: Analyze Statement R
**Statement R**: If D is the midpoint of the hypotenuse AC of a right triangle ABC, then BD = AC.
1. **Draw Right Triangle ABC**: Draw a right triangle ABC with right angle at B.
2. **Identify Midpoint D**: Mark D as the midpoint of hypotenuse AC.
3. **Use Properties of Right Triangles**: In a right triangle, the median to the hypotenuse is half the length of the hypotenuse. Thus, BD = 1/2 AC.
4. **Conclusion**: This statement is false because BD is not equal to AC.
**Result**: False
### Step 4: Analyze Statement S
**Statement S**: The perimeter of a triangle is equal to the sum of its three medians.
1. **Understand Perimeter and Medians**: The perimeter of a triangle is the sum of its three sides, while the sum of the medians is not equal to the perimeter.
2. **Conclusion**: This statement is false.
**Result**: False
### Step 5: Analyze Statement T
**Statement T**: If the altitudes AD, BE, and CF of triangle ABC are equal, then triangle ABC is equilateral.
1. **Draw Triangle ABC**: Draw triangle ABC.
2. **Draw Equal Altitudes**: Draw altitudes AD, BE, and CF such that they are all equal.
3. **Use Properties of Triangles**: If all altitudes are equal, it implies that all sides must also be equal, thus making triangle ABC equilateral.
4. **Conclusion**: This statement is true.
**Result**: True
### Final Summary of Results
- P: True
- Q: True
- R: False
- S: False
- T: True
### Correct Option
The correct option is the one that states: P and Q are true, R and S are false, and T is true.
