The height of the rectangular trapezoid is equal to the smaller base, and one of the corners is equal to 45

| September 15, 2021 | education

The height of the rectangular trapezoid is equal to the smaller base, and one of the corners is equal to 45 degrees. Calculate the length of the middle line of the trapezoid, the length of the smaller base and the length of the segments into which the diagonals of the trapezoid divide the middle line if the larger base of the trapezoid is 8 cm.

Consider a right-angled triangle СDН, in which, according to the condition, the angle СDН = 45, then the angle DСН = 180 – 90 – 45 = 45, therefore, the triangle СDН is rectangular and equilateral, СН = DH.

By condition, the height is CH = BC, and since CH = AB, as heights, then the quadrangle ABCH is a square.

The diagonal AC of the square is also the bisector of the angle VAN, then the angle SAN = CDH = 45, and therefore the triangle ACD is isosceles, and its height CH divides the base AD in half. AH = DH = CH = BC = AB = AD / 2 = 8/2 = 4 cm.

Determine the length of the midline of the trapezoid.

КР = (АD + ВС) / 2 = (8 + 4) / 2 = 12/2 = 6 cm.

In the ABC triangle, the KM segment is its midline and its length is equal to half of the BC base, KM = BC / 2 = 4/2 = 2 cm, similarly, in the ASD triangle, PM = AC / 2 = 8/2 = 4 cm.

Answer: The smaller base is 4 cm, the middle line is 6 cm, the KM segment = 2 cm, PM = 4 cm.



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