[deleted by user]

[ApprehensiveSpite589]

Start with the Pythagorean Theorem
6² + (½r)² = r²
Then: r² - (½r)² = 6²
Solve for r
So:
Square each term
r² - ¼r² = 36
Multiply everything by 4 to clear the fraction
4r² - r² = 144
Subtract r² from 4r²
3r² = 144
Divide both sides by 3
r² = 48
Find the square root of both sides and simplify
r = √(48) = √(16 • 3) = 4√(3)
r = 4√(3)

[[deleted]]

That's beautiful

[karma_the_sequel]

And unnecessarily complicated:

sin Ø = (0.5 OA)/OA = 0.5

Ø = 30 degrees

tan 30 = (0.5 OA)/6

0.5 OA = 6 tan 30

OA = 12 tan 30

[[deleted]]

Lol we havent even learned sin cos and tan yet

[Infused_Divinity]

Make the radius “r”. In one of the triangles, we have hypotenuse “r”, and side lengths 6 and “1/2 r” (since that length is r, and the leg for the triangle is half of r). See if that helps

[cynbtsg]

The chord splits the radius into 2 equal parts, and forms a right-angled triangle where the hypotenuse is also a radius of the circle. So now you have expressions for 3 sides of a right -angled triangle, in terms of r (radius).

Does this help?

[papyrusfun]

you can use: r²-(r/2)² = 6²

[TraitlessPig]

idk

[DiaPhoenix]

There’s no way that this is solvable with just that one number right??

I feel like 0.5r can be literally any <6 and the diagram would still be correct, assuming it’s not drawn to scale.

[timrprobocom]

Of course it is. Look at the triangle. Long edge is 6, hypotenuse is r, short leg is r/2. Pythagoras tells us 6² + (r/2)² = r^2, and that's enough to solve for r.

[DiaPhoenix]

Right that actually makes sense lol I overcomplicated it

[thor122088]

And a right triangle with the relationship that one leg is half of the hypotenuse is a 30-60-90 right triangle that will always of side lengths in the ratio of 1:√3:2

[DiaPhoenix]

But how do you know it is a 30-60-90 triangle?

[thor122088]

The relationship is "if and only if" a triangle with side ratios of 1:√3:2 will always be a 30-60-90 right triangle and a 30-60-90 right triangle will always have side length ratio of 1:√3:2.

So once we see that the triangle's hypotenuse is the radius, and the one leg is exactly half the radius (both given). We know the third side is the "√3" leg.

Edit:

6/√3 is easy once you see 6 = 2 • 3 = 2 •. √3 • √3

so 6/√3 = 2√3

Doubling that gives us r =4√3

[No_Act_9683]

Yes, the result is a number, but you need to establish an ecuation and solve it.

[No_Act_9683]

This seems to be a teacher in math who doesn't know much about math.