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The tangent is perpendicular to the tangent at the point of contact - Leaving Cert Mathematics - Question Question (ii) and (b) - 2012
Question Question (ii) and (b)
The tangent is perpendicular to the tangent at the point of contact. (ii) On the diagram shown, construct the tangent to the circle at A. (b) Construct the circumc... show full transcript
Worked Solution & Example Answer:The tangent is perpendicular to the tangent at the point of contact - Leaving Cert Mathematics - Question Question (ii) and (b) - 2012
Step 1
Construct the tangent at point A
Answer
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Identify Point A: Locate point A on the circle.
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Draw the Radius: Use a straightedge to connect point A to the center of the circle (let's call this point O).
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Draw a Perpendicular Line: At point A, construct a line that is perpendicular to the radius OA. This can be done by ensuring the angle between the radius and the line at point A measures 90 degrees.
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Label the Tangent Line: This line is the tangent to the circle at point A.
Step 2
Construct the circumcentre and circumcircle of the triangle
Answer
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Identify the Triangle Vertices: Label points as A, B, and C for the vertices of the triangle.
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Construct the Perpendicular Bisector of AB: Use a compass to find the midpoint of line segment AB. With the compass, draw arcs above and below the segment from points A and B to create two intersection points. Connect these intersection points with a straightedge to form the perpendicular bisector of AB.
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Construct the Perpendicular Bisector of BC: Repeat the previous step for side BC, creating a perpendicular bisector for this segment as well.
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Identify the Circumcentre O: The point where the two perpendicular bisectors intersect is the circumcentre O of the triangle.
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Draw the Circumcircle: With the compass set to the distance from point O to one of the triangle's vertices (e.g., A), draw a circle around the circumcentre, ensuring that it passes through all three vertices (A, B, C).
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Mark All Construction Points: Clearly indicate the constructed points such as the circumcentre and any key lines used.