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At exactly one point implies the two external lines are tangential. A tangent is perpendicular to the radial line at the point where it touches the circle. Meaning the angle between the tangent line and the radial line is 90° (So ADC = CBA = 90°)
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DC = BC = radius
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in quadrilateral ABCD, all 4 angles should add up to 360°. So we know two angles which are 90° each, so DAB + DCB = 360° - 90° - 90° = 180°
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By symmetry about the axis AC (draw an imaginary line) we can also prove that AD = AB
Thank you @C.Koushik for the clear explanation.
Hi,
Could you please help me understand how to be sure about AD = AB? I got everything else clear, but I want to solidly understand why AD=AB is also true.
If ADAD and ABAB intersect the circle (with CC being the center) at exactly one point, which of the following must be true?
Did you lay out the whole question here? Seems to me the conditions provided in the screenshot here are not sufficient to support all of the options listed except DC=BC
Could you explain this further？
Hey @chloel97
Properties of Tangents
- The tangent line never crosses the circle, it just touches the circle.
- At the point of tangency, it is perpendicular to the radius.
- A chord and tangent form an angle and this angle is same as that of tangent inscribed on the opposite side of the chord.
- From the same external point, the tangent segments to a circle are equal.
I just combined the points 1 and 2 to arrive at what I have stated
Clearly explained here. Hope this helps
Very helpful link! But I was thinking if this case could count,
Since the picture is not drawn to scale, what if point A located like the following?
In this diagram, DA and AB still intersects at one point, which is point A, but clearly they are not tangents to circle.
If a line is not a tangent, then it will intersect the circle in two points
If you extend AB towards right you will see the same, this is why both AB and AD have to be tangents
I don’t seem to quite understand this, can you produce a graph here to illustrate this intersect at 2 points scenario?
Edit: I see what you mean here. And I know where’s the problem. The wording of this question caused confusion. My interpretation of the question is–the two lines DA and AB intersect at one point and thus this point could be everywhere. But you guys interpretation, which is correct, is – AD, AB intersects the circle at one point, respectively,
My bad lol.
If angle CBA is not equal to 90 degrees AB will intersect two points on the circle
If A was at a different point as shown by you in this diagram, point B and D would actually shift, since clearly, if you use a ruler and extend the line AB (from the new point), it would cut into the circle. But the question says the lines don’t cut into the circle (if the line AB were to cut through the circle, it would no longer have a single point of contact, it would have two).
You can try it out yourself to get a better visual context. Keep a cylindrical vessel on a flat surface, and from a point away from the vessel hold a ruler/scale and move it towards the vessel until it touches the vessel. Note the point of contact on the vessel and the ruler. Try the same on the other side of the vessel. You’ll see that the length of the ruler on both sides will be the same when the ruler touches the vessel. Also, try moving your point and repeat the same, the length, the point of contact all changes.
Even though the image is not drawn to scale, it holds that if two lines are extended from the same point to a circle so that the lines only touch the circle at 1 point, the two lines must have the same length.
Nice explanation! I totally understand now. Thanks a lot!
Thank you!