CBSE Class 10 Answered
Theorem: If the tangent to a circle and the radius of the circle intersect they do so at right angles :
Figure (a) Figure (b)
In figure (a) l is a tangent to the circle at A and PA is the radius.
To prove that PA is perpendicular to l , assume that it is not.
Now, with reference to figure (b) drop a perpendicular from P onto l at say B. Let D be a point on l such that B is the midpoint of AD.
In figure (b) consider PDB and PAB
seg.BD congruent to seg.BA ( B is the midpoint of AD)
angle PBD = angle PBA ( PB is perpendicular to l ) and
seg.PB = seg.PB (same segment)
triangle PBD congruent to triangle PAB (SAS)
seg.PD congruent to seg.PA corresponding sides of congruent triangles are congruent.
D is definitely a point on the circle because l (seg.PD) = radius.
D is also on l which is the tangent. Thus l intersects the circle at two distinct points A and D. This contradicts the definition of a tangent.
Hence the assumption that PA is not perpendicular to l is false. Therefore PA is perpendicular to l.
Tangent Secant Theorem: If a chord intersects the tangent at the point of tangency, the angle it forms is half the measure of the intercepted arc. In the figure 1 l is tangent to the circle. Seg.AB which is a chord, intersects it at B which is the point of tangency.
Figure 1
The angles formed angle ABX and angle ABY are half the measures of the arcs they intercept.
angle 1 = m (arc ACB)
angle 2 = m (arc AB)
This can be proved by considering the three following cases.
Figure 2
O is the center of the circle
Arc AXB can be
a) Semi circle
b) minor arc
c) major arc
Case 1 : Assume arc AXB is a semicircle when angle ABC intercepts a semicircle the chord AB passes through the center. Therefore angle ABC = 900 (a tangent is always perpendicular to the diameter that intersects it at the point of tangency).
(arc BXA) = 1800 (arc BXA is a semi circle)
(arc BXA) = ´ 1800 = 900
angle ABC = (arc BXA)
Case 2 : Assume that angle ABC intercepts a minor arc. Therefore as seen in figure 3 the center O lies in the exterior of angle ABC.
Figure 3
angle ABC = 900 - angle ABO
angle ABO = 900 - angle ABC ----------- (1)
But angle ABO = angle OAB
(as OAB is an isosceles triangle )
angle OAB = 900 - angle ABC ----------- (2)
(1) + (2)
angle ABO + angle OAB = 180 - 2 angle ABC
Since angle ABO + angle OAB = 180 - angle BOA
180 - angle BOA = 180 - 2 angle ABC
i.e. angle BOA = 2 angle ABC
angle ABC = angle BOA
angle ABC = ( arc AXB )
Case 3 :
Figure 4
If angle ABC intercepts a major arc, the center of the circle O will lie in the interior as angle ABC . See figure 4.
Now angle ADB intercepts a minor arc AYB.
angle AOB = (arc AYB)
1800 - angle ADB = { 3600 - (arc AXB) }
1800 - angle ADB = 1800 - (arc AXB)
angle ADB = (arc AXB)