In the adjogining figure the radius of aCircle with to Cientere C is 6cm line AB is a tangent at A. Answere the following question
Answers
Answer:
(1) It is given that line AB is tangent to the circle at A.
∴∠CAB=90
o
(Tangent at any point of a circle is perpendicular to the radius throught the point of contact)
Thus, the measure of ∠CAB is 90
o
.
(2) Distance of point C from AB=6cm (Radius of the circle)
(3) △ABC is a right triangle.
CA=6cm and AB=6cm
Using Pythagoras theorem, we have
BC
2
=AB
2
+CA
2
⇒BC=
6
2
+6
2
⇒BC=6
2
cm
thus, d(B,C)=6
2
cm
(4) In right △ABC, AB=CA=6cm
∴∠ACB=∠ABC (Equal sides have equal angles opposite to them)
Also,∠ACB+∠ABC=90
o
(Using angle sum property of triangle)
∴2∠ABC=90
o
⇒∠ABC=
2
90
o
Thus, the measure of ∠ABC is 45
o
Answer:
Join A,C and B,C
( 1 ) We have the theorem,
A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.
∴∠CAB=90
∘
( 2 ) The distance of point C from the line AB is equal to the radius of the circle. ie., line AC
AC=radius of circle=6 cm
( 3 ) AB=6 cm and AC=6 cm
ABC is a right angles triangle.
By Pythagoras theorem,
BC
2
=AB
2
+AC
2
BC
2
=6
2
+6
2
=36+36
BC=
72
=6
2
cm
∴ d(B, C)=6
2
cm
( 4 ) ABC is an isosceles triangle.
AB=AC
⟹∠ABC=∠ACB ------ Angles opposite to equal sides in a triangle are equal
In △ABC,
∠BAC+∠ABC+∠ACB=180
∘
90
∘
+2∠ABC=180
∘
2∠ABC=180
∘
−90
∘
=90
∘
∴∠ABC=
2
90
∘
=45
∘