Three rectangles have exactly the same area. The dimensions of each rectangle (as length and width) are a and b; a – 3 and b + 2; and a + 3 and b – 1. Find the area of the rectangles.
To find the area of each rectangle, we will use the formula: Area = length x width.
Let's start with the first rectangle, which has dimensions a and b. The area of this rectangle is given by A1 = a x b.
For the second rectangle, the dimensions are (a - 3) and (b + 2). The area of this rectangle is A2 = (a - 3) x (b + 2).
Finally, for the third rectangle, the dimensions are (a + 3) and (b - 1). The area of this rectangle is A3 = (a + 3) x (b - 1).
Since all three rectangles have the same area, we can set up the following equation:
A1 = A2 = A3
This means that a x b = (a - 3) x (b + 2) = (a + 3) x (b - 1).
Expanding the equation, we get:
a x b = ab + 2a - 3b - 6 = ab + 3a - b - 3.
Now, let's simplify the equation by canceling out the ab term:
a x b - ab = 2a - 3b - 6 = 3a - b - 3.
Combining like terms, we have:
-ab = 2a - 3b - 6 = 3a - b - 3.
Next, let's isolate one of the variables. We can isolate 'a' by adding 'ab' to both sides of the equation:
0 = 3a + ab - b - 3.
Now, we'll move all terms to one side of the equation:
3a + ab - b - 3 = 0.
Finally, we can solve for 'a' and 'b' by factoring the equation:
3(a + 1) + b(a - 1) = 0.
Now, since the equation equals zero, one of the factors must be zero. This gives us two possibilities:
a + 1 = 0, which gives a = -1, or
a - 1 = 0, which gives a = 1.
Now that we have the values of 'a', we can substitute them back into the previous equation:
If a = -1, then b + 1 = 0, which gives b = -1.
If a = 1, then b - 1 = 0, which gives b = 1.
Therefore, we have two possible sets of values for 'a' and 'b':
Case 1: a = -1, b = -1
Case 2: a = 1, b = 1
For each case, we can calculate the areas of the rectangles:
Case 1:
Area of the first rectangle (A1) = a x b = -1 x -1 = 1.
Area of the second rectangle (A2) = (a - 3) x (b + 2) = (-1 - 3) x (-1 + 2) = -4 x 1 = -4.
Area of the third rectangle (A3) = (a + 3) x (b - 1) = (-1 + 3) x (-1 - 1) = 2 x -2 = -4.
Case 2:
Area of the first rectangle (A1) = a x b = 1 x 1 = 1.
Area of the second rectangle (A2) = (a - 3) x (b + 2) = (1 - 3) x (1 + 2) = -2 x 3 = -6.
Area of the third rectangle (A3) = (a + 3) x (b - 1) = (1 + 3) x (1 - 1) = 4 x 0 = 0.
So, the possible areas of the rectangles are:
- Case 1: A1 = 1, A2 = -4, A3 = -4.
- Case 2: A1 = 1, A2 = -6, A3 = 0.
In summary, the areas of the rectangles, depending on the values of 'a' and 'b', are as follows:
- Case 1: A1 = 1, A2 = -4, A3 = -4.
- Case 2: A1 = 1, A2 = -6, A3 = 0.