Question

select the correct answer from each drop - down menu.
point a lies outside of a circle with center o. the given steps describe the process to start constructing a line tangent to the circle and passing
through point a using a compass and straightedge.
step 1: draw segment \\( \overline { o a } \\).
step 2: find the midpoint, m, of \\( \overline { o a } \\) by constructing the perpendicular bisector of \\( \overline { o a } \\)
complete the missing information for the construction.
step 3: draw a circle centered at
step 4: let the points b and c rep
step 5: draw the segments
a going through m and o
m going through a and o
o going through a and m
reset
next

Explanation:

Step1: Concept of tangent construction

When constructing a tangent from a point outside a circle, one of the key steps is to find the mid - point of the line segment joining the center of the circle and the external point. Then, we draw a circle with this mid - point as the center. The intersection points of this new circle and the original circle give the points of tangency.

Step2: Analyze the steps

In the process of constructing a tangent from an external point \(A\) to a circle with center \(O\):

• We first draw the segment \(\overline{OA}\).
• Then find its mid - point \(M\).
• Next, we draw a circle centered at \(M\). The radius of this circle is \(MA = MO\).
• The two intersection points of the circle centered at \(M\) and the original circle centered at \(O\) are the points of tangency. Let these points be \(B\) and \(C\).
• The lines \(AB\) and \(AC\) are the tangent lines. Since \(M\) is the mid - point of \(OA\) and the circle centered at \(M\) passes through \(O\) and \(A\), the lines \(OB\) and \(OC\) are radii of the original circle and \(\angle OBA=\angle OCA = 90^{\circ}\) (angle in a semi - circle). So \(AB\) and \(AC\) are tangents to the circle centered at \(O\).

Answer:

A going through \(M\) and \(O\)