Question

jenna attempted to construct equilateral triangle efg inscribed in circle c. first, she used a compass to draw circle c and labeled its center. next, she used a straightedge to draw a diameter of the circle. she labeled the endpoints of the diameter d and e. then, keeping the compass set equal to the radius of circle c, she drew an arc centered at point e, which intersected the circle at two points that she labeled f and g. finally, she used a straightedge to draw three chords connecting points e, f, and g. what was the first error jenna made in her construction? a. when jenna labeled the diameter, she should have named the endpoints f and g. b. when jenna drew the arc centered at point e, she should have changed the width of the compass to be less than the radius length. c. when jenna drew the arc that intersected circle c, she should have used point d as the center of the arc. d. when jenna labeled f and g, she should have only labeled f and then drawn a second arc centered at point d to find g. e. when jenna drew the arc centered at point e, she should have changed the width of the compass to be greater than the radius length.

Explanation:

Step1: Analyze the construction of an equilateral triangle inscribed in a circle

When constructing an equilateral triangle inscribed in a circle, after drawing the diameter, the next step is to use the radius length for subsequent arcs. When Jenna drew the arc centered at point \(E\), she should have used the radius length (same as the radius of circle \(C\)) for the compass width. But if she changed the width of the compass (not keeping it equal to the radius), it would be an error.

Step2: Evaluate each option

• Option A: Naming endpoints \(F\) and \(G\) when labeling the diameter is not an error.
• Option B: The arc should be drawn with compass width equal to the radius. If she changed the width to be less than the radius, it is wrong.
• Option C: The center of the arc should be on the circle (either \(D\) or \(E\) when using the diameter endpoints). Using \(D\) as the center for an arc (with radius equal to the circle's radius) is a valid part of the construction.
• Option D: Labeling \(F\) and \(G\) and then using \(D\) as the center for another arc (with radius equal to the circle's radius) is a correct step in some construction methods (e.g., constructing a hexagon - related steps which can lead to an equilateral triangle).
• Option E: Changing the compass width to be greater than the radius is also incorrect, but in the standard construction of an equilateral triangle inscribed in a circle (using the radius for arc - drawing), the first error (when drawing the first non - diameter - related arc) is when the compass width is changed from the radius length.

Answer:

B. When Jenna drew the arc centered at point E, she should have changed the width of the compass to be less than the radius length.