4. Quadrilateral MATH has been dissected into squares. The area of the red square is 64 square units and the area of the blue square is 81 square units. Determine the dimensions and the area of quadrilateral MATH.
To determine the dimensions and area of quadrilateral MATH, we need to understand the relationship between the squares and the quadrilateral.
1. Start by noting that the red square has an area of 64 square units. Since the area of a square is calculated by multiplying its length by its width, we can find the side length (s) of the red square by taking the square root of the area: √64 = 8 units.
Therefore, the dimensions of the red square are 8 units by 8 units.
2. Similarly, the blue square has an area of 81 square units. Using the same calculation, we find that the side length of the blue square is √81 = 9 units.
Therefore, the dimensions of the blue square are 9 units by 9 units.
3. Now, observe that the squares used to dissect the quadrilateral have two of their sides as sides of the quadrilateral. In this case, the red and blue squares share two sides with quadrilateral MATH.
4. Since the red square's sides are 8 units in length and two of these sides align with sides of the quadrilateral, we can say that two sides of the quadrilateral have a length of 8 units each.
5. Similarly, the blue square's sides are 9 units in length, and two of these sides align with sides of the quadrilateral. Hence, the other two sides of the quadrilateral also have a length of 9 units each.
6. Combining these dimensions, we conclude that quadrilateral MATH has two sides of length 8 units and two sides of length 9 units.
7. To calculate the area of quadrilateral MATH, use the formula for the area of a quadrilateral: Area = (Sum of opposite sides)/2 × (Distance between the opposite sides).
8. In this case, the sum of the opposite sides is (8 + 9) units = 17 units, and the distance between the opposite sides is either 8 or 9 units (as the figure is symmetrical). Let's use 9 units as the distance.
9. Therefore, the area of quadrilateral MATH is (17/2) × 9 = 76.5 square units.
In conclusion, the dimensions of quadrilateral MATH are two sides of length 8 units and two sides of length 9 units. The area of quadrilateral MATH is 76.5 square units.