In school, I have an assignment to write a problem for geometry students. I have written the following problem.
Draw triangle ABC. Let the height have magnitude h. Draw a line segment, DE, which is parallel to BC, and which forms trapezoid, DEBC. Let the line segment, DE, have magnitude x, the line segment BC have magnitude b. Let the trapezoid’s height have magnitude y.
Find x and y as functions of b and h.
The assignment further asks me to divide the problem into smaller pieces so that students with less experience might be able to do the problem, so I have modified the problem as follows.
Draw triangle ABC. Let the height have magnitude h. Draw a line segment, DE, which is parallel to BC, and which forms trapezoid, DEBC. Let the line segment, DE, have magnitude x, the line segment BC have magnitude b. Let the trapezoid’s height have magnitude y.
Find the following equations.
To find equation 1 let the area of triangle AED be At. Write At as a function of x, y, and h.
To find equation 2 let the area of trapezoid DEBC be AT. Write AT as a function of x, y, and b.
To find equation 3 let the area of triangle ABC be ATT. Let half of the area of triangle ABC be AT2. Write AT2 as a function of h and b.
To find equation 4 set equation 1 equal to equation 3 and solve for x.
To find equation 5 set equation 2 equal to equation 3 and solve for x.
To find equation 6 set equation 4 equal to equation 5 and solve for y.
Question 6A: Can y be greater than h?
To find equation 7 substitute the result of equation 6 in equation 4 and solve for x.
To find equation 8 substitute the result of equation 7 in equation 5 and solve for y.
To find equation 9 set the value of y of equation 6 equal to the y value of y of equation 8.
Step 10: Use the values of b, h, x, and y to calculate the area of triangle AED.
Step 11 Use the values of b, h, x, and y to calculate the area of trapezoid DEBC.
Step 12: Sum the results of step 10 and step 11.
I would like to know if I have written this properly. Thank you.
Equation 1 is
At = x(h-y)/2
Equation 2 is
AT = (y/2)(x + b)
Equation 3 is
AT2 = bh/4
Equation 4 is
At = x(h-y)/2
AT2 = bh/4
At = AT2
x(h-y)/2 = bh/4
2x(h-y) = bh
2xh – 2xy = bh
x(2h – 2y) = bh
x = bh/(2h – 2y)