My answer will be in the comments section.
Tuesday, June 20, 2023
1 comment:
READ THIS BEFORE COMMENTING!
All comments are subject to approval before they are published, so they will not appear immediately. Comments should be civil, relevant, and substantive. Anonymous comments are not allowed and will be unceremoniously deleted. For more on my comments policy, please see this entry on my other blog.
AND A NEW RULE (per this post): comments critical of Trump's lying must include criticism of Biden's lying on a one-for-one basis! Failure to be balanced means your comment will not be published.
Subscribe to:
Post Comments (Atom)
OK, so we're supposed to find the area of the blue trapezoid. We're given side lengths of
ReplyDelete(x + 1) and (2x + 3) for the horizontals, and
(x + 4) for the vertical, and
(2x + 2) for the diagonal.
So we drop a vertical segment from the upper-left corner of the figure straight down to the bottom leg. From the lower-left corner of the figure to the point of intersection with this new vertical, we know the distance is
(2x + 3) - (x + 1), which simplifies to
x + 2.
We can now solve for x because we have a right triangle with legs of (x + 2) and (x + 4), and a hypotenuse of (2x + 2).
Our Pythagorean equation is therefore
(x+2)^2 + (x+4)^2 = (2x+2)^2
This explodes out to
x^2 + 4x + 4 + x^2 + 8x + 16 = 4x^2 + 8x + 4
This collapses to
2x^2 + 12x + 20 = 4x^2 + 8x + 4
or
2x^2 = 4x + 16, or
x^2 - 2x - 8 = 0,
factoring out to
(x - 4)(x + 2) = 0,
leaving us with two solutions for x:
x = -2, x = 4
We can't use x = -2 because for the side that is (x + 1), we'd get a negative answer, and distances can't be negative. Therefore, x = 4 is the only possible solution.
Knowing that x = 4, we can plug x into the relevant sides to find the area of a trapezoid, which is the height times the average of the two horizontal sides.
height = 8
short horizontal side = 5
long horizontal side = 11
[(5 + 11)/2]•8 = 64.
The trap's area is 64.
QED.