An isosceles trapezoid has a perimeter of 42 inches. Each of the congruent nonparallel sides is 5 inches long and the trapezoid is 3 inches tall. How long are the two parallel sides?
A. 10 in, 22 in
B. 16 in, 16 in
C. 10 in, 16 in
D. 12 in, 20 in
* I understand that you add the sides altogether so 5+5=10 and 42-10=32. Therefore, the sum of the parallel sides equal 32, but I don't know how to find the parallel sides
the 3 and the 5 are memory ticklers
if you draw a 3-4-5 triangle at each end of the longer base
... you'll see that it is 8 inches longer than the shorter base
thank you!
To find the lengths of the two parallel sides of the isosceles trapezoid, you need to use the given information about the nonparallel sides and the height of the trapezoid.
Let's first label the two parallel sides of the trapezoid as "x" inches and "y" inches.
Since both nonparallel sides are congruent and measure 5 inches each, we can find the total length of these two sides by multiplying 5 inches by 2: 2 * 5 = 10 inches.
The perimeter of the trapezoid is given as 42 inches. Therefore, the sum of all four sides is equal to the perimeter. We have the two congruent nonparallel sides with a length of 10 inches each, so their total length is 10 + 10 = 20 inches.
To find the sum of the lengths of the parallel sides: x + y = 20 inches.
Given that the height of the trapezoid is 3 inches, we can represent the slanted sides (the diagonal sides connecting the parallel sides) using the Pythagorean theorem. Since the slanted sides are congruent, we can find their length using:
x^2 = (y - 10/2)^2 + 3^2, where (y - 10/2) represents the distance between the midpoint of one parallel side and the height.
Simplifying this equation, we have:
x^2 = (y - 5)^2 + 9
Now, we know that the perimeter of the trapezoid is equal to the sum of all four sides. Since the length of each nonparallel side is 10 inches and the height is 3 inches, we can set up the equation:
10 + 10 + x + y = 42
Combining like terms, we have:
20 + x + y = 42
Substituting the expression for the sum of the slanted sides into the equation, we get:
20 + (y - 5)^2 + 9 + y = 42
Expanding the equation, we have:
20 + y^2 - 10y + 25 + 9 + y = 42
Combining like terms, we get:
y^2 - 9y + 54 = 0
Factoring the quadratic equation, we have:
(y - 6)(y - 9) = 0
Setting each factor equal to zero, we find two possible values for y:
y - 6 = 0 -> y = 6
y - 9 = 0 -> y = 9
Therefore, the two possible values for the length of the parallel sides are x = 14 and y = 6, or x = 12 and y = 9.
Thus, the correct answer is D. 12 in, 20 in.