In a rectangular trapezoid, the bases are 16 and 25, and the cosine of one of its corners is 0.6.

In a rectangular trapezoid, the bases are 16 and 25, and the cosine of one of its corners is 0.6. Find the smaller diagonal of the trapezoid.

ABCD – rectangular trapezoid
BC = 16
AD = 25
cos alpha = 0.6
CH – trapezoid height

Since the height of the trapezoid is perpendicular to the base, then BC = AH = 16
Then HD = AD – AH
HD = 26 – 16
HD = 9

Consider a triangle CHD, the angle CHD is a straight line, which means a triangle is right-angled
By definition, the cosine of an acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse
Let CH = a, HD = b, CD = c
Then
cos alpha = b / c
0.6 = 9 / s
c = 9 / 0.6
c = 15
CD = c = 15

By the Pythagorean theorem: a ^ 2 + b ^ 2 = c ^ 2
a ^ 2 + 9 ^ 2 = 15 ^ 2
a ^ 2 = 15 ^ 2 – 9 ^ 2
a ^ 2 = (15-9) (15 + 9)
a ^ 2 = 6 * 24
a ^ 2 = 144
a = √144
a = 12
CH = a = 12

Consider a triangle ACH, the angle AHC is a straight line, so the triangle is right-angled. By the Pythagorean theorem:
AC ^ 2 = CH ^ 2 + AH ^ 2
AC ^ 2 = 12 ^ 2 + 16 ^ 2
AC ^ 2 = 144 + 256
AC ^ 2 = 400
AC = √400
AC = 20

Answer: 20