Question

An inscribed triangle with the hypotenuse being the diameter of the circle has angle A be 42 degrees. Angle C is 90 degrees. The area of the triangle ACK is 50cm^2. Find the radius of the circle

Answer 1

Answer:

The Radius of circle is 7.15 cm  

Step-by-step explanation:

Given as :

A Triangle is inscribed into circle

The center of circle is O

The Diameter of circle = AK = d

The hypotenuse of triangle being the diameter of circle

The Area of triangle = 50 cm²

Let The radius of circle = r cm

The Triangle is right angle at c

So , ∠ACK = 90°

∠CAK = 42°

∠AKC = 180° - (90° + 42°)

So, ∠AKC = 48°

Now, ∵ Area of triangle ACK = 50 cm²

So, $\dfrac{1}{2}$ × AC × CK = 50

Or, AC × CK = 50 × 2

i.e , AC × CK = 100        ..........1

From figure

Sin 48° =  $\dfrac{AC}{AK}$

Or, 0.74 =  $\dfrac{AC}{d}$

∴ AC = 0.74 d             ..........2

Similarly

Sin 42° =  $\dfrac{CK}{AK}$

Or, 0.66 =  $\dfrac{CK}{d}$

∴ CK = 0.66 d               .............3

Putting eq 2 and 3 value into eq 1

i.e  AC × CK = 100  

Or, 0.74 d  × 0.66 d = 100  

Or, 0.4884 × d² = 100  

∴ d² = $\dfrac{100}{0.4884}$

Or, d² = 204.75

Or, d = $\sqrt{204.75}$

Or, d = 14.30

So, The diameter of circle = d = 14.30 cm

Now, Radius of circle = $\dfrac{\textrm diameter}{2}$

Or, r =  $\dfrac{\textrm 14.30 cm}{2}$

i.e r = 7.15 cm

So, The Radius of circle =  r = 7.15 cm

Hence, The Radius of circle is 7.15 cm  Answer