Could someone please help me with this?

Question

What are the dimensions of the largest triangle that can be inscribed in a circle of Radius R.

Me and my friend have been working on this one for a few hours, and are completely stumped. Any help would be very wonderful. Thanks!

Answer 1

It will be an equilateral triangle...

You can subdivide the triangle into 6 equal triangles. First draw lines from the center to each vertex. This length is R. Now draw lines from the center to the midpoint of each side to make 6 right triangles. They would each be 30-60-90 triangles, with a hypotenuse of R (the radius of the circle).

Remember the ratio of the sides of 30-60-90 triangle is 1 : 2 : sqrt(3). In your case your triangle has sides R/2 : R : sqrt(3)R/2. The legs are therefore R/2 and sqrt(3)R/2. (You can also use the Pythagorean theorem to derive this result.)

The area of a triangle is (base * height) / 2, so the area of one triangle would be R² sqrt(3) / 8. But you have 6 of them.

So the total area would be:
3/4 R² sqrt(3)

That works out to approximately 1.3 R²

Answer 2

The largest triangle would be an equilateral triangle.

Draw a circle. Label the center O.

Draw a triangle within the circle. Label the three vertices of the triangle A, B, and C.

Draw radii within the circle: OA, OB, and OC.

Now look at the triangles ABO, BCO, and AOC. Because the triangle ABC is equilateral, the triangles ABO, BCO, and AOC have to be congruent, so the size of the inner angle (AOB, BOC or AOC) has to be 360 degrees/3 or 120 degrees.

Look at the triangle ABO. The length of side AO is your radius. The length of side BO is also your radius. The size of the inner angle AOB is 120 degrees.

Can you figure out the length of AB from there?

Answer 3

you will need two items to do this 1. a Hammer and 2. a bowl of warm water


step 1:place the bowl of said water on the florr and kneel over it
step 2: Hit yerself repeatedly with the hammer at which point you should collapse into the bowl and drown.
step 3:if you regain conciousness repeat step 2

Answer 4

Clearly this must be an equilateral triangle.

Each side of such a triangle will be 2r*cos(30).

The height of the triangle will be r + r*sin(30) = 1.5 r

The area of the triangle will be 1.5 r^2 * cos(30).