Kalle draws four shapes on grid paper. Shape \(A\) is a right-angled triangle, shape \(B\) is an isosceles triangle, shape \(C\) is a square, and shape \(D\) is a rectangle. Each shape has a horizontal base and a vertical height.
Using the following clues, determine the base and height for each shape.
The base of shape \(A\) is equal to the base of shape \(D\).
The base of shape \(A\) is one unit less than the base of shape \(B\).
The height of shape \(C\) is equal to the base of shape \(A\).
The height of shape \(B\), the height of shape \(A\), and the base of shape \(B\) are all equal.
The area of shape \(C\) is \(9\) square units.
The total area of all four shapes is \(38\) square units.
First we look at clue \(5\). Since the area of shape \(C\) is \(9\) square units and we know shape \(C\) is a square, then it must have a side length of \(3\) units, since \(3 \times 3 = 9\). Thus, the base and height of shape \(C\) are each \(3\) units.
Then from clue \(3\) we can determine that the base of shape \(A\) is \(3\) units. Then from clue \(1\) we can determine that the base of shape \(D\) is also \(3\) units.
Then from clue \(2\) we can determine that the base of shape \(B\) must be one unit greater than the base of shape \(A\). Thus, the base of shape \(B\) is \(3+1=4\) units.
Then from clue \(4\) we can determine that the height of shape \(B\) and the height of shape \(A\) are also \(4\) units. We now fill in the information we know so far about the base and height of each shape.
Thus, the only information we still need is the height of shape \(D\). We can determine this using clue \(6\). First we will calculate the area of each shape.
Shape \(A\) is a triangle, so its area is \(\text{base} \times \text{height} \div 2 = 3 \times 4 \div 2 = 12 \div 2 = 6\) square units.
Shape \(B\) is a triangle, so its area is \(\text{base} \times \text{height} \div 2 = 4 \times 4 \div 2 = 16 \div 2 = 8\) square units.
Shape \(C\) is a square, so its area is \(\text{base} \times \text{height} = 3 \times 3 = 9\) square units.
Thus, the total area of shapes \(A\), \(B\), and \(C\) is \(6+8+9=23\) square units. Since the total area of all four shapes is \(38\) square meters, it follows that the area of shape \(D\) must be \(38-23=15\) square units.
Shape \(D\) is a rectangle, so its area is \(\text{base} \times \text{height} = 3 \times \text{height} = 15\) square units. It follows that its height must be \(5\) units since \(3 \times 5 = 15\).
The base and height of each shape are as shown.
