complete question :ABCD is a right trapezoid (\angle A = 90°), its diagonals intersect at point G. The circle with diameter AB is tangent to lateral side CD of the trapezoid at point H. Find the length of base BC provided that GH = 5, AD = 7
solution
in a trapezoid AB || CD.
Radius of the circle=7 and diameter=14
i.e. AB=14 and AD=7
We find BC
We will take Δ ADC such that
Angles D and H are 90°
area of triangle ADC=area of triangle CHG and area of trapezium ADHG
21×AD×CD=21×CH×GH+21×(AD+GH)×HD21×7×CD=21×(CD−DH)×5+21×(7+5)×77CD=5CD−5DH+847CD−5CD=−5×7+842CD=49CD=249=24.5
now CD=24.5, BE=AD=7
DE=AB=14
EC=CD-DE
24−1.4=10.5
Now in ΔBEC
BC2=BE2+CE2BC2=72+10.52BC2=159.5BC=12.619
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