The measure of an angle formed by two secants, a secant and a tangent, or two tangents intersecting in the exterior of a circle is equal to one-half the positive difference of the measures of the intercepted arcs.
Problem 1 :
Solution :
x = (1/2) [141 - 61]
x = (1/2) (80)
x = 40
Problem 2 :
Solution :
x = (1/2) [4x + 5 - 50]
x = (1/2) (4x - 45)
Multiplying by 2 on both sides
2x = 4x - 45
2x - 4x = -45
2x = 45
x = 22.5
Problem 3 :
Two secants drawn to a circle from an external point P intercept arcs of 90° and 20° on the circle. Find measure of angle P.
Solution :
Angle p = (1/2)(90 - 20)
= (1/2) (70)
p = 35
Problem 4 :
Solution :
x = (1/2)(82 - 26)
= (1/2) (56)
x = 28
Problem 5 :
Solution :
28 = (1/2)( 3x - 4 - (x + 4) )
28 = (1/2)(3x - 4 - x - 4)
28 = (1/2)(2x - 8)
28 = (1/2) 2(x - 4)
28 = x - 4
28 + 4 = x
x = 32
Problem 6 :
Solution :
40 = (1/2)(x - 80)
Multiplying by 2 on both sides
80 = x - 80
Adding 80 on both sides.
x = 80 + 80
x = 160
Problem 7 :
Solution :
Since P is the center, the line passing through the center will divide the circle into two equal parts. Hence it is semi circle.
Let t be the angle measure intercepted by the smaller arc.
148 + 25 + t = 180
173 + t = 180
t = 180 - 173
t = 7
x = (1/2) (25 - 7)
x = (1/2)(18)
x = 9
Problem 8 :
Solution :
46 = (1/2)(134 - x)
Multiplying by 2 on both sides.
46(2) = 134 - x
92 = 134 - x
Subtracting 134 and adding x on both sides.
92 - 134 = -x
x = 42
Problem 9 :
Solution :
42 = (1/2)(x - 66)
Multiplying by 2 on both sides.
42(2) = x - 66
84 = x - 66
Adding 66 on both sides.
x = 84 + 66
x = 150
Problem 10 :
Solution :
30 = (1/2)( 2x - 30 - (x + 15) )
30 = (1/2)(2x - 30 - x - 15)
30 = (1/2)(x - 45)
Multiplying by 2 on both sides.
60 = x - 45
x = 60 + 45
x = 105
Problem 11 :
Solution :
55 = (1/2)(4x - y)
Multiplying by 2 on both sides.
110 = 4x - y
y = 4x - 110 -----(1)
2x + 4x + y = 360
6x + y = 360 -----(2)
Applying (1) in (2), we get
6x + 4x - 110 = 360
10x - 110 = 360
add 110 on both sides
10x = 360 + 110
10x = 470
x = 47
Applying the value of x in (1), we get
y = 4(47) - 110
y = 188 - 110
y = 78
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM