Question 1: In a circle, a radius is drawn from the center to a point on the circle. Construct a line through the point that is perpendicular to the radius. What is the name of this line?

Answer: Tangent

Question 2: Use the grid to draw a radius from the center of a circle to a point on the circumference. Construct a perpendicular line at that point. Verify if the line is perpendicular using a right angle marker or grid.

Answer: Perpendicular at contact point

Question 3: True or False: The line drawn from the center of the circle to the point where a tangent touches the circle is always perpendicular to the tangent line.

Answer: True

Question 4: On the grid, draw a circle with center O. Mark a point P on the circle. Draw radius OP. Now, construct the tangent line at P. Show that it is perpendicular to OP.

Answer: Perpendicular at P

Question 1: A circle has radius 5 cm. A radius is drawn from the center to point P on the circle. A tangent line is drawn at P. If a point Q outside the circle is 8 cm from the center, explain how you can determine whether Q is along the tangent line at P.

Answer: Use the perpendicular radius at P and check the distance from Q to the tangent line or circle center.

Question 2: Given a circle with center O and radius 6 cm, a radius OP is drawn to point P on the circle. A tangent line at P is constructed. If a point R is 4 cm from O, how can you determine if R lies on the tangent line at P?

Answer: Calculate the distance from R to the tangent line and compare with perpendicular distance.

Question 3: Explain why the tangent line at P is always perpendicular to the radius OP, using a geometric proof or reasoning.

Answer: Because the tangent line at P is the unique line that touches the circle at P without crossing, and the radius creates a right angle with the tangent at P due to congruent triangles or circle properties.

Question 4: In a diagram, a radius is drawn from the center to point P on the circle. The student constructs a line through P that appears parallel to the radius. Identify and explain the common misconception in this construction.

Answer: Students might confuse tangent lines with parallel lines; the tangent is perpendicular, not parallel, to the radius.

Question 1: A car tire is modeled as a circle. The radius from the axle to the point where the tire touches the ground is 30 cm. Explain how technicians use the property of perpendicularity of the tangent to check for proper tire contact and alignment.

Answer: Technicians ensure the contact patch is tangent at the point of contact, with the radius perpendicular to the ground, to verify proper alignment.

Question 1: Construct a circle with radius 4 cm. From a point outside the circle, draw two tangents to the circle. Prove that the angles between the radii to the points of contact and the tangent lines are right angles, using geometric reasoning.

Answer: Angles are right angles because the tangent lines are perpendicular to the radii at the points of contact.

Question 2: Given a circle with radius 7 cm, if a tangent at point P is drawn and a point Q is outside the circle, write the steps to find the shortest distance from Q to the tangent line, explaining how the perpendicular radius helps in this calculation.

Answer: Drop a perpendicular from Q to the tangent; use the property that the radius at the contact point is perpendicular to the tangent to assist in setting up the calculation.

Question 1: A student draws a radius and then a line through the point on the circle, claiming it is the tangent because it looks perpendicular. The student marks a small box at the right angle. The student then proceeds to draw the tangent line. Identify the mistake and explain the correct procedure.

Answer: The student assumed the line is tangent without constructing it accurately. The correct method is to construct the tangent as a line perpendicular to the radius at P, using a right angle tool or geometric construction.

Question 2: A student claims that any line through point P on the circle and the center O is perpendicular to the radius. Is this correct? Explain why or why not.

Answer: No, only the line constructed as the tangent at P is perpendicular to the radius at P.

Question 3: A student draws a radius and a line through the contact point that is not perpendicular. Explain a possible reason for this mistake and how to avoid it.

Answer: The student may not have used a right angle tool or failed to construct the perpendicular correctly; to avoid, they should reliably use geometric tools to construct the perpendicular.