In a rectangular trapezoid ABCD (AD || BC), the diagonal AC is perpendicular to the CD side. Find the length of this diagonal if AD = 18 cm, BC = 8 cm.
Let’s draw the height CH of the trapezoid ABCD.
Quadrilateral ABCH is a rectangle, then AH = BC = 8 cm, then DH = (AD – AH) = (18 – 8) = 10 cm.
Triangle ACD is rectangular by condition, then by the property of height, lowered from the vertex of the right angle to the hypotenuse: CH ^ 2 = AH * DH.
CH ^ 2 = 8 * 10 = 80.
CH = √80 = 4 * √5 cm.
From the right-angled triangle ACH, we determine, by the Pythagorean theorem, the length of the hypotenuse AC.
AC ^ 2 = AH ^ 2 + CH ^ 2 = 64 + 80 = 144.
AC = 12 cm.
Answer: The length of the hypotenuse is 12 cm.
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