The trapezoid diagonal ABCD (AD || BC) are mutually perpendicular and intersect at point O.

| September 5, 2021 | education

The trapezoid diagonal ABCD (AD || BC) are mutually perpendicular and intersect at point O. Find the side CD if it is known that BO = 9, OD = 12, AC = 11⅔.

Consider triangles BCO and AOD, in which the angles BOC and AOD are equal since AC is perpendicular to BD, and the angles BCO and ADO are equal as criss-crossing angles at the intersection of parallel lines AD and BC of the secant BD, therefore the triangles BCO and AO are similar in two angles.

Let the segment OC = X cm, then the segment OA = 11 (2/3) – X = 35/3 – X.

Then:

BO / DO = CO / AO.

9/12 = X / (35/3 – X).

12 * X = 105 – 9 * X.

21 * X = 105.

X = 5 cm.

CO = 5 cm.

Consider a right-angled triangle СОD and, by the Pythagorean theorem, find the hypotenuse СD.

CD ^ 2 = OC ^ 2 + OD ^ 2 = 25 + 144 = 169.

СD = 13 cm.

Answer: Side CD = 13 cm.



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