**Applying properties of chord examples :**

Here we are going to see example problems based on applying properties of chord.

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

<AOB = <DOC

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

AC = BC

Equal chords of a circle are equidistant from the centre.

OM = OL

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.

m<1 = (1/2) (Measure arcCD+measure of arcAB)

m<2 = (1/2) (Measure arcBC+measure of arcCD)

If two chords intersect inside a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

EA ⋅ EB = EC ⋅ ED

**Example 1 :**

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.

**Example 2 :**

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°.

**Example 3 :**

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

**Example 4 :**

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.

**Example 5 :**

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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