Final answer:
To calculate the length of MN, we use the information provided about the rectangle and the right-angled triangle. We start by finding the combined length of the remaining sides of the rectangle. We then set up an equation using the area formula of the triangle and the fact that NO = 6 cm - MN. Solving this equation will give us the length of MN.
Step-by-step explanation:
In order to calculate the length of MN, we need to use the information given regarding the rectangle and the right-angled triangle. Firstly, we know that the perimeter of the rectangle is 30 cm, and the length of LM is 12 cm. Since a rectangle has opposite sides of equal length, the width of the rectangle is also 12 cm. Therefore, the total length of all four sides of the rectangle is 2(12 cm) + 2(12 cm) = 24 cm. Since the perimeter of the rectangle is given as 30 cm, this means that the combined length of the remaining two sides, OP and NO, must be 30 cm - 24 cm = 6 cm.
Next, we are informed that MNO is a right-angled triangle with an area of 12 cm². The formula for the area of a triangle is A = 1/2 * base * height. In this triangle, MN is the base and NO is the height. Therefore, we can set up the equation 1/2 * MN * NO = 12 cm².
Using the information that the combined length of OP and NO is 6 cm, we can substitute NO with 6 cm - MN to get a simplified equation: 1/2 * MN * (6 cm - MN) = 12 cm². Solving this equation will give us the length of MN.