Question 1066301
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AB is the diameter of a circle. AD and BC are tangents to the circle with AD = 9cm and BC = 16cm. 
If AC and BD intersect at a point on the circle, then the length, in centimetres, of AB is:
a) 5.76 cm  b) 9 cm  c) 12 cm  d) 12.5 cm  e) 25 cm
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<pre>
0.  Make a sketch.

    Let "x" be the length of AB, which is under the question: x = |AB|.


1.  Let P be the intersection point of the segments AC and BD. 

    As the condition says, the point P lies on the circle.
    Therefore, the angle APB is the right angle (it leans on the diameter AB !)

    Thus, the segments AC and BD are perpendicular.

    Also, let's denote a = |BP|, b = |AP|.


2.  Right-angled triangles ABC and APB are similar (they have the common acute angle BAP).
    Therefore, their corresponding sides are proportional: {{{abs(BC)/abs(AB)}}} = {{{abs(BP)/abs(AP)}}},  or  {{{16/x}}} = {{{a/c}}}.    (1)


3.  Right-angled triangles BAD and BPA are similar (they have the common acute angle ABP).
    Therefore, their corresponding sides are proportional: {{{abs(AD)/abs(AB)}}} = {{{abs(AP)/abs(BP)}}},  or  {{{9/x}}} = {{{c/a}}}.    (2)


4.  Divide (1) by (2) (both sides). You will get {{{(a/c)^2}}} = {{{16/9}}}, which implies {{{a/c}}} = {{{sqrt(16/9)}}} = {{{4/3}}}.


5.  Now substitute {{{4/3}}} instead of {{{a/c}}} into (1).

    You will get {{{16/x}}} = {{{4/3}}}, which implies x = {{{(16*3)/4}}} = 12.
</pre>

<U>Answer</U>.  &nbsp;|AB| = 12 cm.     &nbsp;&nbsp;&nbsp;&nbsp;Option c).