Question

Given:

KLMN is a trapezoid, KL=MN,

AB is a midsegment, AB=18,

KL=15, LM:KN=1:5

Find the height.

Answer 1

9 units

Explanation:

By the midsegment theorem,

`AB=(KN+LM)/(2)`

Since the ratio
`LM:KN=1:5,` then
`LM=x,\ KN=5x` and

`18=(x+5x)/(2)\\ \\18=(6x)/(2)\\ \\18=3x\\ \\x=6\\ \\LM=6\ units\\ \\KN=30\ units`

Draw two heights LE and MF. Quadrilateral ELMF is a rectangle, then EF = LM = 6 units.

Trapezoid KLMN is isosceles trapezoid (KL = MN), then KE = FN and

`KE=FN=(KN-EF)/(2)=(30-6)/(2)=12\ units`

Consider right triangle KLE. By the Pythagorean theorem,

`KL^2=LE^2+KE^2\\ \\15^2=LE^2+12^2\\ \\LE^2=15^2-12^2\\ \\LE^2=225-144=81\\ \\LE=9\ units`