Question

select the correct answer. a student constructs a line tangent to circle o at point p on the circle. she followed the given steps. step 1: draw a line from the center o through p that extends beyond the circle. step 2: with a compass on p, draw an arc on each side of p and label q and r. step 3: with a compass on q, set its width equal to qp and make an arc. step 4: with the compass on r, using the same width as the previous step, make another arc intersecting the previous arc and label the intersection s. step 5: draw a line through p and s. which step contains an error and how can it be fixed? a. step 2: set the compass on o, not p. b. step 5: draw a line through o and s. c. step 4: set the width of the compass equal to op. d. step 3: set the width greater than qp.

Explanation:

Step1: Analyze Step 2

The correct way to start constructing the tangent is to use the point of tangency \(P\) as the center for the first - arc drawing with the compass. So, setting the compass on \(O\) (as in option A) is incorrect.

Step2: Analyze Step 5

The tangent line should be drawn through the point of tangency \(P\) and the intersection point \(S\). Drawing a line through \(O\) and \(S\) (as in option B) will not give the tangent line.

Step3: Analyze Step 4

The width of the compass in step 4 should be the same as the width used in step 3 (equal to \(QP\)) to construct the perpendicular bisector - like structure for the tangent. Setting it equal to \(OP\) (as in option C) is incorrect.

Step4: Analyze Step 3

The width of the compass in step 3 should be set equal to \(QP\) to create congruent arcs for the construction of the perpendicular at \(P\). Setting it greater than \(QP\) (as in option D) will disrupt the construction of the perpendicular at \(P\) which is required for the tangent line.

The correct way to construct a tangent to a circle at a point \(P\) on the circle is to first draw the radius \(OP\), then construct a perpendicular to \(OP\) at \(P\). In the given steps, step 3 has an error. When constructing the arcs to find the point \(S\) which will help in drawing the perpendicular (and thus the tangent), the width of the compass in step 3 should be set equal to \(QP\).

Answer:

D. Step 3: Set the width equal to \(QP\) (the original statement in option D is wrong as it says set the width greater than \(QP\))