Lines of Symmetry for an Equilateral Triangle and Circumcircle
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Exercise 5.1
$\triangle ABC$ is an equilateral triangle and circle O is its circumcircle. How many lines of symmetry does figure 5.5 have?
Figure 5.5
This question includes visual content: Figure 5.5 shows a circle with center point O. Inside the circle, an equilateral triangle ABC is inscribed. The vertices A, B, and C lie exactly on the circle's circumference. Vertex C is at the top, and side AB is horizontal at the bottom. The center O of the circle is also the centroid/center of the triangle.
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Step by Step Written Solution
In this problem, we are looking at an equilateral triangle inscribed in a circle, and we want to determine how many lines of symmetry this figure has.
Symmetry of Figure 5.5
Let's start by reconstructing the figure with the equilateral triangle A B C and its circumcircle with center O.
A line of symmetry is a line that divides a figure into two mirror-image halves. Since the circle itself has infinite lines of symmetry through the center O, the symmetry of the whole figure is determined by the triangle.
Key Properties:
- Circle: Infinite symmetry
- Equilateral Triangle: 3 lines of symmetry
Let's draw the first line of symmetry. For an equilateral triangle, this passes through a vertex and the midpoint of the opposite side.
The first line passes through vertex C and the center O. Notice how the left side is a perfect mirror of the right side.
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