Question

select the correct answer. casey constructed this figure by using a compass and a straightedge to draw circle o and diameter $overline{pq}$. she then used a compass to draw two arcs. she labeled the points where those arcs intersect the circle m and n, and she drew the chords that form the triangle. if $\triangle mnp$ is equilateral, how did casey draw the arcs that defined points m and n? a. she used a compass centered at p with a radius set equal to pq. b. she used a compass centered at o with a radius set equal to po. c. she used a compass centered at q with a radius set equal to po. d. she used a compass centered at p with a radius set equal to $\frac{3}{4}(pq)$. e. she used a compass centered at q with a radius set equal to pq.

Explanation:

Step1: Recall properties of equilateral - triangle in a circle

In a circle, if we want to construct an equilateral triangle with one vertex on the diameter - end point. When we use a compass centered at one end - point of the diameter (say \(P\)) and set the radius equal to the diameter \(PQ\), we can get the other two vertices of the equilateral triangle on the circle.
Let the radius of the circle be \(r\), and the diameter \(PQ = 2r\). If we consider the geometric construction of an equilateral triangle inscribed in a circle with one side along the diameter, using the compass centered at one end - point of the diameter with radius equal to the diameter length will give the correct construction.

Step2: Analyze each option

• Option A: If we use a compass centered at \(P\) with a radius set equal to \(PQ\), we can construct an equilateral triangle \(\triangle MNP\) inscribed in the circle.
• Option B: Using a compass centered at \(O\) with radius equal to \(PO\) will just give points on the circle at a distance equal to the radius from the center, not an equilateral - triangle construction.
• Option C: Using a compass centered at \(Q\) with radius equal to \(PO\) will not give the correct construction for an equilateral triangle with one vertex at \(P\).
• Option D: Using a compass centered at \(P\) with radius equal to \(\frac{3}{4}(PQ)\) will not result in an equilateral triangle.
• Option E: Using a compass centered at \(Q\) with radius equal to \(PQ\) will not give the correct construction for an equilateral triangle with one vertex at \(P\).

Answer:

A. She used a compass centered at \(P\) with a radius set equal to \(PQ\).