Radius of a circle is 10 cm and distance of a chord from the centre is 6 cm hence, the length of the chord is___ *
Answers
Answer -
- Radius of a circle is 10 cm and distance of a chord from the centre is 6 cm hence, the length of the chord is 16 cm.
Given :
- Radius of a circle = 10 cm
- Distance of the chord from the centre = 6 cm
To find :
- Length of the chord
Concept :
Here, the concept of circle theormes and pythagoras theorem will be used.
Circle theorem to be used :-
- Perpendicular drawn from the centre bisects the chord, i.e. divides the chord into two equal parts.
Pythagoras theorem :-
In a right angled triangle, the square of the Hypotenuse is equal to the sum of the squares of the other two sides, i.e. perpendicular and the base.
Mathematically,
- H² = P² + B²
where,
- H = Hypotenuse [Longest side]
- P = Perpendicular
- B = Base
Solution :
In ∆AOP
By using pythagoras theorem,
- H² = P² + B²
we have,
- H = AO = 10 cm
- P = OP = 6 cm
- B = AP = ?
→ (10)² = (6)² + (AP)²
→ Transposing (6)² to the left hand side. On transposing it's sign will also be changed.
→ (10)² - (6)² = AP²
→ Using algebraic identity,
- (a - b)(a + b) = a² - b²
→ (10 - 6)(10 + 6) = AP²
→ (4)(16) = AP²
→ Taking square root on both the sides.
→ √(4 × 16) = AP
→ √(2 × 2 × 4 × 4) = AP
→ ± 2 × 4 = AP
→ ± 8 = AP
→ As we know, side of the triangle cannot be in negative. So, the negative sign will get rejected.
→ ± 8 Reject - ve = AP
→ 8 = AP
Therefore,
- Base of the triangle (AP) = 8 cm
Using circle theorem,
→ Length of the chord (AB) = 2 × AP
→ AB = 2 × 8
→ AB = 16
Therefore,
- Length of the chord (AB) = 16 cm
