Ø What is the effect on the area of a triangle if the base is doubled and the height is cut in half? What happens to the area, if the base is doubled and the height remains the same?
a = 1/2 bh
1/2 (2b)(h/2) = 1/2 bh = a -- unchanged
1/2 (2b)(h) = bh = 2a -- area is doubled
To understand the effect of changing the base and height of a triangle on its area, we need to know the formula for calculating the area of a triangle. The formula for the area of a triangle is given as:
Area = (base * height) / 2
Now, let's analyze the effects of the changes mentioned in the questions.
1) If the base is doubled and the height is cut in half:
To find the new area, we substitute the new values into the formula:
New area = (2 * base * (1/2 * height)) / 2
Simplifying this expression, we get:
New area = (base * height) / 2
Notice that the expression for the new area is equal to the original formula for the area of a triangle. Therefore, doubling the base and halving the height has no effect on the area of the triangle. The area remains the same.
2) If the base is doubled and the height remains the same:
Similarly, to find the new area, we substitute the new values into the formula:
New area = (2 * base * height) / 2
Simplifying this expression, we get:
New area = (base * height) / 1
Here, the expression for the new area is just the product of the base and height, which is double the original formula. Therefore, if we double the base while keeping the height constant, the area of the triangle is also doubled.
In summary:
- Doubling the base and halving the height has no effect on the area of the triangle; it remains the same.
- Doubling the base while keeping the height constant will double the area of the triangle.