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The given triangle \( ABC \) has midpoints \( P \), \( T \), and \( S \) marked on sides \( AB \), \( AC \), and \( BC \) respectively. We are provided with the lengths of:

- \( AC = 70 \, \text{mm} \)
- \( BC = 84 \, \text{mm} \)

### Midpoint Theorem:
According to the Midpoint Theorem, the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

### Steps to calculate:

1. **For \( PS \) (joining midpoints \( P \) and \( S \)):**
   - Since \( P \) and \( S \) are midpoints of sides \( AB \) and \( BC \), \( PS \) is parallel to \( AC \) and half its length.
   - Thus, \( PS = \frac{1}{2} \times AC = \frac{1}{2} \times 70 = 35 \, \text{mm} \).

2. **For \( PT \) (joining midpoints \( P \) and \( T \)):**
   - Since \( P \) and \( T \) are midpoints of sides \( AB \) and \( AC \), \( PT \) is parallel to \( BC \) and half its length.
   - Thus, \( PT = \frac{1}{2} \times BC = \frac{1}{2} \times 84 = 42 \, \text{mm} \).

### Final Answer:
- \( PS = 35 \, \text{mm} \)
- \( PT = 42 \, \text{mm} \)

Would you like more details on the Midpoint Theorem or any other part of this solution?

### 5 Related Questions:
1. How does the Midpoint Theorem help in simplifying triangle problems?
2. What are some other properties of midpoints in triangles?
3. How do you prove that \( PS \parallel AC \) and \( PT \parallel BC \)?
4. Can the Midpoint Theorem be applied in quadrilaterals, and if so, how?
5. How would the calculation change if only the perimeter of \( \triangle ABC \) were given?

### Tip:
Whenever solving geometric problems, always check if midpoints or parallel sides are involved, as they often lead to useful theorems like the Midpoint Theorem or properties of similar triangles.

Given triangle ABC with midpoints P, T, and S on sides AB, AC, and BC respectively, and given that AC = 70 mm and BC = 84 mm, calculate the lengths of PS and PT.

Learn how to calculate the lengths of PS and PT in a triangle where P, T, and S are midpoints, with side lengths AC = 70 mm and BC = 84 mm. This problem applies the Midpoint Theorem to simplify the solution, suitable for students in Grades 7-9.

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