Which of the following statements is false
A. Every chord is part of a secant
B. Every diameter is part of a secant
C. Every chord is a diameter
D. Every diameter is a chord
An equilateral triangle with sides of length 6 is inscribed in a circle. What is the diameter of the circle?
A. 5.2
B. 6
C. 6.9
D. 7.5
C. Every chord is a diameter is false.
Okay what about
An equilateral triangle with sides of length 6 is inscribed in a circle. What is the diameter of the circle?
A. 5.2
B. 6
C. 6.9
D. 7.5
We can use the fact that the circumcenter of an equilateral triangle is the intersection point of its perpendicular bisectors. Since the triangle has sides of length 6, its altitude is $\sqrt{6^2 - 3^2} = 3\sqrt{3}$. The perpendicular bisector of each side passes through the circumcenter, and since the triangle is equilateral, we know that these bisectors are also the altitudes of the triangle. Therefore, the distance from the circumcenter to any side is $3\sqrt{3}$, which is also the radius of the circle. Therefore, the diameter is $2 \cdot 3\sqrt{3} = 6\sqrt{3} \approx \boxed{6.9}$.
Can someone dumb down what it said? I only see #&@^*&^&@#^*^&%*^*.
Sure! Essentially, the question is asking for the length of a segment of a circle (its diameter) when given a triangle inscribed in that circle. The triangle is an equilateral triangle, meaning all of its sides are equal, and these sides have a length of 6.
To find the diameter of the circle, we use the fact that the circumcenter of an equilateral triangle is the point where the perpendicular bisectors of the sides intersect. In other words, if we draw a line that is perpendicular to a side of the triangle and cuts it in half, and do this for all three sides, these lines will meet at a single point. That point is the circumcenter.
From there, we find that the distance from the circumcenter to any side of the triangle (which is also the radius of the circle) is $3\sqrt{3}$, which means the diameter of the circle is $2\cdot 3\sqrt{3} = 6\sqrt{3}$. This is about 6.9 when rounded to one decimal place, so the answer is (C) 6.9.