Find the area of the triangle defined by the points \( (2,3),(-4,3) \) and \( (-6,6) \). 9 16 18 4.5 QUESTION 3 Which shapes have the same area? \[ \text { I. }(1,1),(1,4),(4,4),(4,1) \] II. \( (0,0),(0,-4.5),(-2,0),(-2,-4.5) \) III. \( (4,-3),(2,-5),(6,-5) \) They all have different areas II and III I, II and III I and II

To find the area of the triangle formed by the points \( (2,3),(-4,3) \), and \( (-6,6) \), we can use the formula for the area of a triangle given by three points \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\): \[ \text{Area} = \frac{1}{2} | x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) | \] Substituting in the coordinates, we get: \[ \text{Area} = \frac{1}{2} | 2(3-6) + (-4)(6-3) + (-6)(3-3) | \] Calculating this gives: \[ \text{Area} = \frac{1}{2} | 2(-3) + (-4)(3) + 0 | = \frac{1}{2} | -6 - 12 | = \frac{1}{2} \times 18 = 9 \] Thus, the area of the triangle is \( 9 \). --- Now, for the shapes with the same area: Calculating the areas, we find: 1. Shape I is a square with side length 3, leading to an area of \( 3 \times 3 = 9 \). 2. Shape II is a rectangle with dimensions \( 2 \times 4.5 = 9 \). 3. Shape III is a triangle with a base of 4 (from points (4,-3) to (0,-3)) and a height of 2, resulting in an area of \( \frac{1}{2} \times 4 \times 2 = 4 \). So, shapes I and II both have the same area of 9, while shape III has an area of 4. Therefore, the correct answer is: I and II.