Let's analyze the question step by step. Here's the information given:

Points:

• $A(3, 4)$
• $B(6, 7)$
• $C(9, 4)$
• $D(7, 2)$

We are asked:

a) Name the type of quadrilateral formed by joining points A, B, C, and D in order.

b) Find the coordinate of the midpoint of AC.
OR
Find the coordinate of the midpoint of BD.

c) Find the distance between the midpoints of AC and BD.

Let's break this down:

a) Type of Quadrilateral:

To determine the type of quadrilateral, we will check the lengths of the sides and diagonals, and check if the angles between them are right angles.

Points:

• $A(3, 4)$, $B(6, 7)$, $C(9, 4)$, $D(7, 2)$

We calculate the lengths of all sides using the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• AB: $\sqrt{(6-3)^2 + (7-4)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$
• BC: $\sqrt{(9-6)^2 + (4-7)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$
• CD: $\sqrt{(7-9)^2 + (2-4)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$
• DA: $\sqrt{(7-3)^2 + (2-4)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$

Based on this, the sides do not seem to all be equal, and thus it could be a trapezium.

b) Midpoint of AC and BD:

The midpoint formula is: $\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

• Midpoint of AC: $\left( \frac{3+9}{2}, \frac{4+4}{2} \right) = (6, 4)$
• Midpoint of BD: $\left( \frac{6+7}{2}, \frac{7+2}{2} \right) = \left( \frac{13}{2}, \frac{9}{2} \right) = (6.5, 4.5)$

c) Distance between midpoints:

We will use the distance formula again to find the distance between the midpoints of AC and BD: $\text{Distance} = \sqrt{(6.5 - 6)^2 + (4.5 - 4)^2} = \sqrt{(0.5)^2 + (0.5)^2} = \sqrt{0.25 + 0.25} = \sqrt{0.5} \approx 0.707$

Would you like more details on any specific part of this? Here are some related questions for further exploration:

1. How do you classify other types of quadrilaterals?
2. How does changing one coordinate affect the shape of a quadrilateral?
3. Can the midpoints be used to identify special quadrilateral properties?
4. What would the result be if the coordinates formed a rectangle or square?
5. How would you verify if the quadrilateral is cyclic?

Tip: Always verify the classification of a quadrilateral by checking both side lengths and angles.