Question

lesson 4
practice problems
9 problems
1 this diagram is a straightedge and compass construction. a is the center of one circle, and b is the center of the other.
explain how we know triangle abc is equilateral.
2 a, b, and c are the centers of the three circles. how many equilateral triangles are there in this diagram?
3 this diagram is a straightedge and compass construction. a is the center of one circle, and b is the center of the other.
select all the true statements.
a ac = bc
b ac = bd
c cd = ab
d quadrilateral abcd is a square.
e triangle abd is an equilateral triangle.
f cd = ab + ab
geometry • unit 1 • section a • lesson 4

Explanation:

Step1: Recall circle - radius property

In a circle, all radii are equal. In the first problem, for the circle with center \(A\), \(AC = AB\) (radii of circle \(A\)). For the circle with center \(B\), \(BC=AB\) (radii of circle \(B\)). So \(AC = AB=BC\). By the definition of an equilateral triangle (a triangle with all sides equal), \(\triangle ABC\) is equilateral.

Step2: Analyze the second - diagram

In the second diagram, consider the properties of circles and equilateral triangles. Triangles \(\triangle ABC\), \(\triangle ACH\), \(\triangle BCF\), \(\triangle ADE\), \(\triangle BDE\) are equilateral. The reason is that the distances between the centers of the circles and the intersection - points are equal to the radii of the circles. For example, in \(\triangle ABC\), \(AB = BC=CA\) (radii of the circles centered at \(A\), \(B\), and \(C\)). Counting them, there are 5 equilateral triangles.

Step3: Analyze the third - problem

In the third diagram, since \(A\) and \(B\) are the centers of the circles:

• \(AC\) and \(BC\) are radii of the two circles. \(AC = BC\) because the two circles are constructed in such a way that the radii are equal, so option A is true.
• \(AC\) and \(BD\) are radii of the two circles, so \(AC = BD\), option B is true.
• \(CD\) and \(AB\) are not equal. The length of \(CD\) is not the same as the length of \(AB\), so option C is false.
• Quadrilateral \(ABCD\) is a rhombus (all sides \(AC = BC=BD = AD\) are radii of the two circles), but not a square because the angles are not right - angles, so option D is false.
• \(\triangle ABD\) is not an equilateral triangle. The lengths \(AD\) and \(BD\) are radii, but the shape does not satisfy the equilateral - triangle condition, so option E is false.
• \(CD

eq AB + AB\), so option F is false.

Answer:

1. Since \(AC = AB\) (radii of circle \(A\)) and \(BC = AB\) (radii of circle \(B\)), \(AC=AB = BC\), so \(\triangle ABC\) is equilateral.
2. 5
3. A. \(AC = BC\), B. \(AC = BD\)