Inside the isosceles trapezoid ABCD with bases BC = 12, AD = 28 and the lateral side CD = 10, point O

| March 17, 2021 | education

Inside the isosceles trapezoid ABCD with bases BC = 12, AD = 28 and the lateral side CD = 10, point O is chosen so that the circle centered at point O touches the bases of the trapezoid and the side CD. Find the area of triangle ABO.

Let’s build the height of the HВ. Since the trapezoid is isosceles, then AH = (AD – BC) / 2 = (28 – 12) / 2 = 8 cm.Then, by the Pythagorean theorem, BH ^ 2 = AB ^ 2 – AH ^ 2 = 100 – 64 = 36 …

BH = 6 cm.

Point O lies on the midline of the trapezoid. Let’s construct a segment OM.

In a right-angled triangle ВKM, ВM = AB / 2 = 5 cm, ВK = ВН / 2 = 3 cm, ВK = AH / 2 = 3 cm.

By the property of the trapezium, into which the circle is inscribed, the triangle СОD is rectangular, in which OP is the median drawn to the hypotenuse, then OP = CD / 2 = 10/2 = 5 cm.

Then OM = MР – OР = (AB + BC) / 2 – OR = 20 – 5 = 15 cm.

The AOB triangle consists of two triangles, OBM and OAM, which have a common side OM and the height of both triangles is 3 cm.Then Som = Soam = OM * ВK / 2 = 15 * 3/2 = 22.5 cm2.

Then Saov = 2 * 22.5 = 45 cm2.

Answer: The area of ​​the AOB triangle is 45 cm2.



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