SOLUTION: If the lengths of the bases of an isosceles trapezoid inscribed in a circle are 10 cm and 22 cm, and if one of the legs is 10 cm, then what is the length of a diagonal, in cm?

Algebra ->  Points-lines-and-rays -> SOLUTION: If the lengths of the bases of an isosceles trapezoid inscribed in a circle are 10 cm and 22 cm, and if one of the legs is 10 cm, then what is the length of a diagonal, in cm?      Log On


   

Question 1150801: If the lengths of the bases of an isosceles trapezoid inscribed in a circle are 10 cm and 22 cm, and if one of the legs is 10 cm, then what is the length of a diagonal, in cm?
Answer by MathLover1(20850) About Me  (Show Source):
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Make a sketch.


Let ABCD be the given isosceles trapezoid.
The bases are AB = 22 cm long;  CD = 10 cm;
the lateral sides are AD = BC = 10 cm.






Draw the perpendiculars CE and DF from the vertices C  and D to the base AB.

It is clear that the right-angled triangles ADF  and BCE are congruent.

Then the segments AF and BE have the length  %28AB+-+CD%29%2F2 = %2822-10%29%2F2 = 12%2F2 = 6 cm each.


The height of the trapezoid CE is the leg of the right angled triangle BCE and its length is equal to  sqrt%2810%5E2-6%5E2%29 = sqrt%28100-36%29 = sqrt%2864%29 = 8 cm.


Now from the right-angled triangle AEC you have

    AC = sqrt%28%2810%2B6%29%5E2%2B8%5E2%29 = sqrt%2816%5E2%2B8%5E2%29 = sqrt%28256%2B64%29 = sqrt%28320%29 = sqrt%2816%2A20%29 = 4%2Asqrt%2820%29 = 8%2Asqrt%285%29.


Thus the diagonal of the trapezoid AC is  8%2Asqrt%285%29 cm long.   ANSWER