A right-angle triangle is stretched so that the base of the triangle is half as wide, but the hypotenuse is twice the length of the original. The area of the resulting triangle will be:

Question

larger than the original.

the same size as the original.

smaller than the original.

exactly half the original.

Answer 1

The area of a triangle is calculated using the formula A = 1/2 * base * height.

In the original right-angle triangle, let the base be 'b' and the height (perpendicular to the base) be 'h'. The hypotenuse can be calculated using the Pythagorean theorem: hypotenuse = sqrt(b^2 + h^2).

If the base of the triangle is stretched to half its original width, the new base will be 1/2 * b = b/2.
If the hypotenuse is stretched to twice its original length, the new hypotenuse will be 2 * hypotenuse = 2 * sqrt(b^2 + h^2).

In the resulting triangle, the area can be calculated using the same formula: A' = 1/2 * (b/2) * height'.

Comparing the areas of the original triangle (A) and the resulting triangle (A'), we need to compare the expressions:

A = 1/2 * b * h
A' = 1/2 * (b/2) * height'

Let's analyze each option:
1. Larger than the original: A' > A
2. The same size as the original: A' = A
3. Smaller than the original: A' < A
4. Exactly half the original: A' = 1/2 * A, so A' could be smaller or larger.

Now, let's substitute the height in terms of b and h using the Pythagorean theorem:
A' = 1/2 * (b/2) * height'
= 1/2 * (b/2) * sqrt((b/2)^2 + h^2)
= (b^2 / 8) * sqrt(1 + 4(h/b)^2)

Comparing A' and A, it is clear that A' is smaller than A:

A' = (b^2 / 8) * sqrt(1 + 4(h/b)^2) < 1/2 * b * h = A

Therefore, the resulting triangle has a smaller area than the original triangle. Hence, the answer is "smaller than the original".