Answer: Without a perpendicular cut, we cannot determine the height of the triangle directly. However, we can use the Pythagorean Theorem to find it indirectly.
Let's assume that Francine has cut the paper into a right triangle ABC, where AB and BC are the legs of the triangle, and AC is the hypotenuse. Let x be the length of AB and y be the length of BC. We can then use the Pythagorean Theorem to find the length of AC:
AC^2 = AB^2 + BC^2
We don't know the exact values of AB and BC, but we do know that the area of the triangle is given by:
Area = (1/2) * AB * BC
We can use this formula to find the area in terms of x and y:
Area = (1/2) * x * y
Now we need to eliminate one of the variables (either x or y) so that we can express the area in terms of the other variable and AC. We can do this by rearranging the Pythagorean Theorem to solve for one of the variables:
x^2 = AC^2 - y^2
y^2 = AC^2 - x^2
Let's solve for y:
y^2 = AC^2 - x^2
y = sqrt(AC^2 - x^2)
Now we can substitute this expression for y into the formula for the area:
Area = (1/2) * x * sqrt(AC^2 - x^2)
To find the maximum possible area, we need to maximize this expression. We can do this using calculus, by taking the derivative of the expression with respect to x, setting it equal to zero, and solving for x. However, since we only need an approximate answer to the nearest tenth of a square centimeter, we can use trial and error to find the value of x that gives the maximum area. We can start by trying different values of x, and using a calculator to evaluate the expression and find the corresponding area.
Let's try a few values of x:
If x = 0, then the area is 0.
If x = AC/2 (i.e., if the triangle is isosceles), then the area is (1/8) * AC^2 * sqrt(3), which is approximately 0.433 * AC^2.
If x = AC/sqrt(3) (i.e., if the triangle is equilateral), then the area is (1/4) * AC^2 * sqrt(3), which is approximately 0.433 * AC^2.
Since the area is maximized when the triangle is equilateral, we can assume that the piece of paper is an equilateral triangle, and use the formula for the area of an equilateral triangle:
Area = (sqrt(3)/4) * side^2
where side is the length of one of the sides of the triangle. To find the side length, we need to know the length of AC. We don't know the exact value of AC, but we can estimate it by measuring the length of the longest side of the piece of paper (assuming that it is the hypotenuse of the triangle). Let's say that the longest side is 30 cm. Then, by the Pythagorean Theorem, we can find the length of the other two sides:
x^2 + y^2 = 30^2
x^2 + (sqrt(AC^2 - x^2))^2 = 30^2
2x^2 + AC^2 - x^2 = 900
x^2 = (900 - AC^2)/
Explanation: