Question
Prove that any three points on a circle cannot be collinear.
✓
Answer
To Prove: seg CP ≅seg CQ
Construction: Join CP, CQ and CT
Figure:
Since PQ is a tangent to the circle, ∠CTP = ∠CTQ = 90
Since ∠APT = ∠CTP = 90
AP || CT.
Similarly,
CT || BQ.
So, we can say that,
AP || CT || BQ
AB is a line cutting all three parallel lines.
AC = CB (Radius of the circle, and AB is diameter, C is center)
Since C is center point of line AB cutting parallel lines.
We can say these parallel lines are equal distance.
Therefore, PT = TQ.
Now in ΔCTP and ΔCTQ,
CT is a common side, PT = TQ and ∠CTP = ∠CTQ = 90
CP = CQ. (Pythagoras theorem or congruent triangle theorem)
Hence, Proved.
Construction: Join CP, CQ and CT
Figure:
Since PQ is a tangent to the circle, ∠CTP = ∠CTQ = 90
Since ∠APT = ∠CTP = 90
AP || CT.
Similarly,
CT || BQ.
So, we can say that,
AP || CT || BQ
AB is a line cutting all three parallel lines.
AC = CB (Radius of the circle, and AB is diameter, C is center)
Since C is center point of line AB cutting parallel lines.
We can say these parallel lines are equal distance.
Therefore, PT = TQ.
Now in ΔCTP and ΔCTQ,
CT is a common side, PT = TQ and ∠CTP = ∠CTQ = 90
CP = CQ. (Pythagoras theorem or congruent triangle theorem)
Hence, Proved.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
A circle with centre $P$ is inscribed in the $\triangle ABC.$ Side $AB,$ side $BC$ and side $AC$ touches the circle at points $L, M$ and $N$ respectively. Radius of the circle is $r$
Prove that : $A (\triangle ABC )=\frac{1}{2}(A B+B C+A C) \times r$
→Prove that : $A (\triangle ABC )=\frac{1}{2}(A B+B C+A C) \times r$
From a cubical piece of wood of side $21\ cm,$ a hemisphere is carved out in such a way that the diameter of the hemisphere is equal to the side of the cubical piece. Find the surface area and volume of the remaining piece.
→Solve$:\sqrt{x^2-16}-\sqrt{x^2-8 x+16}=\sqrt{x^2-5 x+4}$
→The areas of two similar triangles are $81\ cm^2$ and $49\ cm^2$ respectively. If the altitudes of the first triangle is $6.3\ cm,$ find the corresponding altitude of the other.
→$3 x+y=1 ; 2 x=11 y+3$
→If a cone of radius $10\ cm$ is divided into two parts by drawing a plane through the mid-point of its axis, parallel to its base. Compare the volumes of the two parts.
→One year ago, a man was $8$ times as old as his son. Now, his age is equal to the square of his son's age. Find their present ages.
→One tank can be filled up by two taps in 6 hours. The smaller tap alone takes 5 hours more than the bigger tap alone. Find the time required by each tap to fill the tank separately.
Draw a circle of radius 3cm. Take two points P and Q on one of its extended diameter each at a distance of 7cm from its centre. Draw tangents to the circle from these two points P and Q.
A cylindrical bucket of height $32\ cm$ and base radius $18\ cm$ is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed.
If the height of the concial heap is $24\ cm$, find the radius and slant height of the heap.
→If the height of the concial heap is $24\ cm$, find the radius and slant height of the heap.