Question

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 . step 2: find the midpoint, m, of by constructing the perpendicular bisector of . complete the missing information for the construction. step 3: draw a circle centered at . step 4: let the points b and c represent the points where the two circles meet. step 5: draw the segments and to create two tangent lines to the circle.

Answer 1

The construction of tangent lines from point A to a circle with center O involves drawing a circle with midpoint M of segment OA and connecting the two intersection points B and C with point A to form the tangents AB and AC.

Step-by-step explanation:

The student's question is related to the construction of tangent lines to a circle using a compass and straightedge. To complete the construction process after finding the midpoint M of segment OA, where O is the center of the circle and A is the external point:

1. Draw a circle centered at M that passes through O and intersects the original circle at two points; let's call these points B and C.
2. Draw segments AB and AC. These segments are the required tangent lines to the original circle from point A since they touch the circle exactly at points B and C, respectively.

By the tangent radius theorem, we know that a tangent to a circle is perpendicular to the radius at the point of tangency. Hence, lines AB and AC, both tangent to the circle at B and C, will be perpendicular to the radii OB and OC, respectively.

Answer 2

1. segment ao
2. midpoint of ao; bisector of ao
3. center: m
4. -
5. segments ab, ac

Step-by-step explanation:

You want to complete the steps to draw tangents ab and ac from point a external to circle o.

Step 1:

Draw segment ao .

Step 2:

Find the midpoint, m, of ao by constructing the perpendicular bisector of ao .

The missing information for the bisector construction:

• Using the same radius larger than ma, draw intersecting arcs centered at point a and point o on either side of segment ao.
• Join the points of intersection of those arcs with the segment bisecting ao.
• The point where the segments cross is point m.

Step 3:

Draw a circle centered at m with radius ma.

Step 4:

Let the points b and c represent the points where the two circles (circle m and circle o) meet.

Step 5:

Draw the segments ab and ac to create two tangent lines to the circle.