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:
a. exactly half the original.
larger than the original.
b. smaller than the original.
d. the same size as the original.
The area of a right triangle is given by the formula A = (1/2) * base * height.
Let's assume that the original triangle has a base of length b and a height of length h. The original area is then A1 = (1/2) * b * h.
After the triangle is stretched, the base becomes (1/2) * b and the hypotenuse becomes 2h.
The height of the stretched triangle can be found using the Pythagorean theorem: (1/2) * b^2 + h^2 = (2h)^2.
Simplifying this equation yields:
(1/4) * b^2 + h^2 = 4h^2.
Multiplying through by 4 gives:
b^2 + 4h^2 = 16h^2.
Subtracting 4h^2 from both sides gives:
b^2 = 12h^2.
The area of the stretched triangle is then A2 = (1/2) * (1/2) * b * 2h = h * b.
We can now compare A2 with A1:
A2/A1 = (h * b) / ((1/2) * b * h) = 2.
Therefore, the area of the resulting triangle is b. smaller than the original. Answer: b. smaller than the original.