Question

What is the radius of the circle

Answer 1

Check the picture below.

so its radius is simply half of its diameter, so let's find the diameter.

`\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2\implies c=√(a^2 + o^2) \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{AB}\\ a=\stackrel{adjacent}{9}\\ o=\stackrel{opposite}{12} \end{cases} \\\\\\ AB=√( 9^2 + 12^2)\implies AB=√( 81 + 144 ) \implies AB=√( 225 )\implies AB=15 \\\\\\ radius=\cfrac{15}{2}\implies radius = 7.5`

Answer 2

7.5

Explanation:

Thales Theorem states that if A, B, and C are points on the circumference of a circle, where AB is the diameter, then the angle at point C is a right angle.

Therefore, since triangle ABC is a right triangle, we can use Pythagoras Theorem to find the length of the hypotenuse (AB).

`\boxed{\begin{array}{l}\underline{\sf Pythagoras \;Theorem} \\\\\Large\text{$a^2+b^2=c^2$}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a$ and $b$ are the legs of the right triangle.}\\\phantom{ww}\bullet\;\textsf{$c$ is the hypotenuse (longest side) of the right triangle.}\\\end{array}}`

From observation of the triangle:

• a = BC = 9
• b = AC = 12
• c = AB

Substitute these values into the formula and solve for AB:

`\begin{aligned}9^2+12^2&=AB^2\\81+144&=AB^2\\AB^2&=225\\AB&=√(225)\\AB&=15\end{aligned}`

Therefore, the length of AB is 15 units.

Since the radius (r) of a circle is half its diameter, and AB is the diameter of circle O, then:

`r=(AB)/(2)=(15)/(2)=7.5\; \sf units`

Therefore, the radius of circle O is:

`\Large\boxed{7.5}\; \sf units}`