Given: KLMN is a trapezoid, KL = MN, AB is a midsegment, AB = 18, KL = 15, LM : KN = 1 : 5
Find: The height of the trapezoid.
Let's denote the height of the trapezoid as h, and the lengths of the bases KN and LM as x and 5x, respectively.
Since AB is a midsegment, we have AB = (KN + LM)/2 = (x + 5x)/2 = 3x = 18. Thus, x = 6.
Using the relationship from the given ratios: LM : KN = 1 : 5, we have LM = 1/6*KM and KN = 5/6*KM.
Since KL = 15, we have KM = KL - LM = 15 - (1/6)*15 = 15 - 2.5 = 12.5
Now, let's consider the right triangle KLM. We can use the Pythagorean theorem to find the height h:
h^2 + (LM)^2 = (KN)^2
h^2 + (1/36)*(KM)^2 = (5/36)*(KM)^2
h^2 = (5/36)*(KM)^2 - (1/36)*(KM)^2
h^2 = (4/36)*(KM)^2
h^2 = (1/9)*(12.5)^2
h^2 = (1/9)*156.25
h^2 = 17.36 (rounded to 2 decimal places)
Therefore, the height of the trapezoid is approximately 4.17 units.