**Applying properties of chords worksheet :**

Here we are going to see some practice questions on applying properties of chords.

(1) Find the value of "y". | |

(2) Find the value of "x". | |

(3) Find the value of "x". | |

(4) Find the value of "x". | |

(5) Find the value of "x". | |

(6) Find the value of "x". | |

(7) Find the value of "x". | |

(8) Find the value of "x". |

**Question 1 :**

Find the value of "y".

**Solution :**

A perpendicular drawn from the center of the circle, bisects the chord.

AE = EB = 15/2 = 7.5 cm

In triangle OEB,

OB^{2} = OE^{2} + EB^{2}

OB^{2} = 7^{2} + (7.5)^{2}

OB^{2} = 49 + 56.25

OB^{2} = 105.25

OB = √105.25

OB = 10.25 cm

OB = OD = radius of the circle = 10.25 cm

In triangle OFD,

OD^{2} = OF^{2} + FD^{2}

(10.25)^{2} = 7^{2} + FD^{2}

(10.25)^{2} - 7^{2} = FD^{2}

FD^{2} = 105.06 - 49

FD^{2} = 56.06

FD = √56.06

FD = 7.48 cm

CD = 2(7.48) = 14.96 cm

Hence the value of y is 14.96 cm.

**Question 2 :**

Find the value of x.

**Solution :**

80° = (1/2)[Measure of AB + Measure of CD]

80° = (1/2)(x + 60°)

80 (2) = x + 60

160 - 60 = x

x = 100°

Hence the value of x is 100°.

**Question 3 :**

Find the value of x.

**Solution :**

<ACE = <BCD

x + 63 = 3x + 1

x - 3x = 1 - 63

-2x = -62

x = 62/2

x = 31

Hence the value of x is 31.

**Question 4 :**

Find the value of "x".

**Solution :**

**Property going to be used :**

Equal chords of a circle subtend equal angles at the centre.

Since AE = BD,

<ACE = <BCD

2x - 5 = x

2x - x = 5

x = 5

Hence the value of x is 5.

**Question 5 :**

Find the value of "x".

**Property going to be used :**

If two chords intersect inside a circle, then the measure of each angle formed is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

x = (1/2) [measure of arc AB + measure of arc DC]

x = (1/2) [80 + 40]

x = (1/2) (120)

x = 60°

Hence the value of x is 60°.

**Question 6 :**

Find the values of "x".

**Property going to be used :**

Perpendicular from the centre of a circle to a chord bisects the chord.

AC = BC = 4 cm

In triangle OCB,

OB^{2} = OC^{2} + BC^{2}

x^{2} = 3^{2} + 4^{2}

x^{2} = 9 + 16

x^{2} = 25

x = √25

x = 5 cm

**Question 7 :**

Find the values of "x".

**Solution :**

**EH **⋅ HG = JH ⋅ HF

**4 **⋅ 10 = 8 ⋅ x

x = (**4 **⋅ 10) / 8

x = 40/8

x = 5 cm

Hence the value of x is 5 cm.

**Question 8 :**

AB is a diameter of the circle below. If BC = 2 m and AB = 9 m, find the
exact length of AC .

**Solution :**

In triangle ABC,

<BCA = 90°

AB^{2} = AC^{2} + BC^{2}

9^{2} = AC^{2} + 2^{2}

81 - 4 = AC^{2}

AC = √77

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