A right triangle has a base, b, that is 6 inches. The area of the triangle, with h representing the height, is given by the expression bh/2. Complete the table to show how the area of the triangle changes with height: Height:4, Area:12, height:5, area:15, height:6, area:18, height:7, area:21, height:8, area:24. How does the area change as the height increases? Why do you think this happens?
Answer (1)
To understand how the area of a right triangle changes with its height, let's break it down step-by-step.
The area A of a triangle is given by the formula:
A = 2 b × h
In this case, the base b is 6 inches. Therefore, the expression for the area becomes:
A = 2 6 × h = 3 h
Now, we'll look at how the area changes as the height h increases:
Height: 4 inches
Area: 3 × 4 = 12 square inches
Height: 5 inches
Area: 3 × 5 = 15 square inches
Height: 6 inches
Area: 3 × 6 = 18 square inches
Height: 7 inches
Area: 3 × 7 = 21 square inches
Height: 8 inches
Area: 3 × 8 = 24 square inches
How does the area change with height?
As we can see from the calculations above, the area increases by 3 square inches every time the height increases by 1 inch.
Why does this happen?
This linear relationship occurs because the area is directly proportional to the height when the base is constant. In the formula A = 3 h , the area A increases linearly with the height h since the base 6 is a constant multiplier. This means for every increase of 1 inch in height, the area increases by 3 × 1 = 3 square inches efficiently, leading to a consistent pattern.
