To solve the given statements, we will analyze each statement one by one and determine whether they are true (T) or false (F).
### Statement (i):
**If the points A(6,1), B(8,2), C(9,4) and (p,3) are the vertices of a parallelogram, taken in order, then the value of p is 7.**
**Solution:**
1. **Identify the coordinates of the points:**
- A(6, 1)
- B(8, 2)
- C(9, 4)
- D(p, 3)
2. **Use the property of a parallelogram:**
- The diagonals of a parallelogram bisect each other. Therefore, the midpoint of AC should be equal to the midpoint of BD.
3. **Calculate the midpoints:**
- Midpoint of AC = ((6 + 9)/2, (1 + 4)/2) = (15/2, 5/2)
- Midpoint of BD = ((8 + p)/2, (2 + 3)/2) = ((8 + p)/2, 5/2)
4. **Set the midpoints equal to each other:**
- (15/2, 5/2) = ((8 + p)/2, 5/2)
5. **Equate the x-coordinates:**
- 15/2 = (8 + p)/2
- Multiply both sides by 2:
- 15 = 8 + p
- Rearranging gives:
- p = 15 - 8 = 7
**Conclusion for Statement (i):**
The value of p is indeed 7. Therefore, this statement is **True (T)**.
### Statement (ii):
**The points A(2,-2), B(-3,8) and C(-1,4) are collinear.**
**Solution:**
1. **Use the collinearity condition:**
- Three points A(x1, y1), B(x2, y2), C(x3, y3) are collinear if the area formed by them is zero.
- The area can be calculated using the determinant:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
2. **Substituting the coordinates:**
- A(2, -2), B(-3, 8), C(-1, 4)
\[
\text{Area} = \frac{1}{2} \left| 2(8 - 4) + (-3)(4 + 2) + (-1)(-2 - 8) \right|
\]
\[
= \frac{1}{2} \left| 2(4) - 3(6) + 1(10) \right|
\]
\[
= \frac{1}{2} \left| 8 - 18 + 10 \right| = \frac{1}{2} \left| 0 \right| = 0
\]
**Conclusion for Statement (ii):**
Since the area is zero, the points are collinear. Therefore, this statement is **True (T)**.
### Statement (iii):
**The points A(-3,0), B(1,-3) and C(5,1) are the vertices of an isosceles right angled triangle.**
**Solution:**
1. **Calculate the lengths of the sides using the distance formula:**
- Distance AB:
\[
AB = \sqrt{(1 - (-3))^2 + (-3 - 0)^2} = \sqrt{(1 + 3)^2 + (-3)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]
- Distance BC:
\[
BC = \sqrt{(5 - 1)^2 + (1 - (-3))^2} = \sqrt{(4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]
- Distance CA:
\[
CA = \sqrt{(5 - (-3))^2 + (1 - 0)^2} = \sqrt{(8)^2 + (1)^2} = \sqrt{64 + 1} = \sqrt{65}
\]
2. **Check for isosceles right triangle:**
- For a triangle to be an isosceles right triangle, two sides must be equal and the square of the longest side must equal the sum of the squares of the other two sides.
- Here, the sides are 5, 4√2, and √65.
- Check if any two sides are equal: 5 ≠ 4√2 and 5 ≠ √65, and 4√2 ≠ √65.
- Therefore, the triangle is not isosceles.
**Conclusion for Statement (iii):**
The points do not form an isosceles right triangle. Therefore, this statement is **False (F)**.
### Final Answers:
- (i) T
- (ii) T
- (iii) F
