Question

select the correct answer.
a student constructed tangent lines to circle o, passing through point p outside the circle. he followed the given steps.
step 1: draw a segment from the center o to p.
step 2: construct the perpendicular - bisector of segment op. label the mid - point m.
step 3: set the compass width to be greater than mo.
step 4: from point m, draw an arc across the circle and label the intersections n and q.
step 5: draw a line from n to p and another line from q to p.
which step contains an error and how can it be fixed?
step 5: draw a line through m and p and m and q
step 4: from point o draw an arc across the circle and label n
step 2: construct a bisector of segment op
step 3: set the width equal to mo.

Explanation:

Step1: Analyze step - by - step construction

When constructing tangent lines from an external point $P$ to a circle with center $O$, the correct steps involve finding the mid - point $M$ of $OP$, then using $M$ as the center and $MO$ (or $MP$ since $M$ is the mid - point of $OP$) as the radius to draw a circle that intersects the original circle at the points of tangency.

Step2: Check step 3

In step 3, setting the compass width to be greater than $MO$ is incorrect. The correct width of the compass should be set equal to $MO$ (or $MP$). When we set the compass width equal to $MO$ and draw an arc from $M$, the intersection points of the arc and the circle are the points of tangency.

Step3: Check other steps

Step 1 is correct as we start by connecting the center of the circle $O$ to the external point $P$. Step 2 is correct as constructing the perpendicular bisector of $OP$ gives us the mid - point $M$ which is the center of the circle that will help us find the points of tangency. Step 4 is incorrect in the original construction as it is based on the wrong compass - width setting in step 3. Step 5 is correct in the sense that the lines from the intersection points on the circle to the external point $P$ are the tangent lines but it is based on wrong previous steps.

Answer:

Step 3: Set the width equal to $MO$