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11 Feb 2020, 01:17
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C
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42% (02:54) correct
58%
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wrong
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In the figure given below, O is the center and AB is the diameter of the circle of radius 15. From point D, two tangents DC and DB are drawn to the circle. If AC is parallel to OD and AC+OD=43, find the length of CD given CD is positive integer
A. 16
B. 18
C. 20
D. 22
E. 24
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11 Feb 2020, 04:42
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When I read this question, the first thing that struck me as odd was that they mentioned "CD is a positive integer" (which presumably means that the length of CD is a positive integer)
Length would obviously be positive but needn't be an integer. If it is specified and seeing that OCD is a right triangle (tangent makes 90 degree angle with radius), this might have something to do with pythagorean triplets.
The ones with 15 or factors of 15 (since OC = 15) are
3, 4, 5 (the sides can be 15, 20, 25)
5, 12, 13 (the sides can be 15, 36, 39)
8, 15, 17 (the sides can be 8, 15, 17)
(OC is one leg and CD is the other)
The only length given in the options is 20 (we have neither 36 nor 8)
Answer (C)
Length would obviously be positive but needn't be an integer. If it is specified and seeing that OCD is a right triangle (tangent makes 90 degree angle with radius), this might have something to do with pythagorean triplets.
The ones with 15 or factors of 15 (since OC = 15) are
3, 4, 5 (the sides can be 15, 20, 25)
5, 12, 13 (the sides can be 15, 36, 39)
8, 15, 17 (the sides can be 8, 15, 17)
(OC is one leg and CD is the other)
The only length given in the options is 20 (we have neither 36 nor 8)
Answer (C)
11 Feb 2020, 15:11
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Since AC || OD, < CAO = < COD.
—> if you draw a segment CB, then triangle ACB will be right-angled triangle.
—> Also, triangle ACB and triangle OCD are similar triangles.
—> \(\frac{AB}{AC} = \frac{OD}{OC}\)
\(\frac{30}{AC} = \frac{OD}{15}\)
AC*OD = 450
AC+ OD = 43
—> AC= 18, OD = 25
(Using Pythagorean theorem)
\(CD^{2} + OC^{2} = OD^{2}\)
—> \(CD^{2} = 625 —225 = 400\)
CD = 20
The answer is C
Posted from my mobile device
—> if you draw a segment CB, then triangle ACB will be right-angled triangle.
—> Also, triangle ACB and triangle OCD are similar triangles.
—> \(\frac{AB}{AC} = \frac{OD}{OC}\)
\(\frac{30}{AC} = \frac{OD}{15}\)
AC*OD = 450
AC+ OD = 43
—> AC= 18, OD = 25
(Using Pythagorean theorem)
\(CD^{2} + OC^{2} = OD^{2}\)
—> \(CD^{2} = 625 —225 = 400\)
CD = 20
The answer is C
Posted from my mobile device
