Answer:
- The line AOC is diameter of the circle with centre O.
- Angle ABC is right angle i.e. 90⁰ .
- Area AMB is arc of circle with centre O.
- The radius of circle with centre O is 5 units .
- The area of ∆ ABC is 24 sq. units.
Step-by-step explanation:
PART 1 :
- Line AOC is a straight line which passes through centre and intersect the circle at two points A and C.
- Hence, it is diameter of the circle with centre O.
PART 2 :
- The inscribed angle theorem states that: " An angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle. "
- Angle AOC subtends 180⁰ at the centre of the circle and angle ABC is half of the angle subtended at the centre.
- Angle ABC is :
[tex] \frac{ {180}^{0} }{2} = {90}^{0} [/tex]
- Hence, Angle ABC is a right angle.
PART 3 :
- Area AMB is the arc of the circle with centre o.
PART 4:
- From geometry of the figure ABC, it is evident that ∆ABC is a right angled triangle with right angle at B.
- By Pythagoras theorem:
(AB)² + (BC)² = (AC)²
- Given that AB = 8 cm and BC = 6 cm.
- This gives AC =
[tex] \: \: \: \: \: \: \: \: = \sqrt{ {8}^{2} + {6}^{2} } \\ \: \: \: \: \: \: \: = \sqrt{ 64 + 36 } \\ = \sqrt{100} \\ = 10 \: cm[/tex]
- AC is the diameter of the circle with centre O. Radius of the circle is
[tex] \frac{10}{2} = 5 \: cm[/tex]
PART 5 :
- ∆ ABC is a right angled triangle with right angle at B.
- Base of triangle = 6 cm
- Height of triangle = 8 cm
- Area of triangle is given by :
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \frac{1}{2} \times base \times height \\ = \frac{1}{2} \times 8 \times 6 \\ = 24 \: sq \: cm[/tex]
Hence, area of ∆ ABC is 24 sq. cm .