The angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle.

**Example 1 :**

Find the value of x in the following figure.

**Solution :**

According to the theorem,

∠BOC = 2∠BAC -----(1)

∠BOC = 2x

∠BOC + ∠BOA + ∠AOC = 360°

∠BOC + 120 + 90 = 360

∠BOC + 210 = 360

∠BOC = 360 - 210

∠BOC = 150°

From this we may find the value of ∠BAC. That is x.

From (1)

∠BOC = 2∠BAC

150 = 2∠BAC

∠BAC = 150/2

= 75°

So, the value of x is 75°.

**Example 2 :**

Find the value of x in the following figure.

**Solution :**

The angle in a semicircle is right angle.

∠ACB = 90°

Sum of interior angels in a triangle is 180°.

∠CAB + ∠ACB + ∠ABC = 180

x + 90 + 35 = 180

x + 125 = 180

x = 180 - 125

x = 55°

**Example 3 :**

Find the value of x in the following figure.

OA = OB = OC ( radius )

∠OCA = ∠OAC = 30°

∠OBA = ∠OAB = 25°

∠BOC = 2∠BAC

∠BAC = ∠BAO + ∠OAC

= 25 + 30

= 55°

∠BOC = 2(55) = 110°

So, the value of x is 110°.

**Example 4 :**

Find the value of x in the following figure.

**Solution :**

reflex ∠AOC = 360 - 130 = 230°

2∠ABC = reflex of ∠AOC

Reflex of ∠AOC = 2x

x = Reflex of ∠AOC / 2

= 230/2

= 115°

So, the value of x is 115°.

**Example 5 :**

Find the value of x in the following figure.

**Solution :**

In triangle DBC,

∠DBC + ∠DCB + ∠CDB = 180°

∠CDB = 90°

50 + x + 90 = 180°

140 + x = 180°

x = 180 - 140

x = 40°

So, the value of x is 40°.

**Example 6 :**

Find the value of x in the following figure.

∠AOB = 2∠ACB

∠AOB = 2(48) = 96

In triangle AOB,

∠OAB + ∠ABO + ∠BOA = 180

OA = OB = OC = Radius

∠OAB = ∠OBA = x

x + x + 96 = 180

2x + 96 = 180

2x = 180 - 96

2x = 84

x = 84/2

x = 42°

So, the value of x is 42°.

**Example 7 :**

Find the value of x in the following figure.

**Solution :**

Using the theorem the angle subtended by an arc of a circle at the centre is double the angle subtended by it at any point on the remaining part of the circle.

(i) ∠AOB = 2∠ACB

∠ACB = (1/2)∠AOB

= (1/2) ⋅ 80°

= 40°

**Example 8 :**

Find the value of x in the following figure.

**Solution :**

reflex ∠AOB = 2∠ACB

x = 2 ⋅ 100° = 200°

**Example 9 :**

Find the value of x in the following figure.

∠ABC + ∠BCA + ∠CAB = 180°

56° + 90° + ∠CAB = 180°

(∠BCA = angle on a semicircle = 90°)

∠CAB = 180° - 146°

x = 34°

**Example 10 :**

Find the value of x in the following figure.

OA = OB = OC ( radius )

∠OCA = ∠OAC = 25°

∠OBC = ∠OCB = 20°

∠ACB = ∠OCA + ∠OCB

= 25° + 20° = 45°

AOB = 2∠ACB

x = 2(45°)

x = 90°

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

**v4formath@gmail.com**

We always appreciate your feedback.

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**