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If a line is tangent to a circle, then the line is perpendicular to the radius drawn from the center to the point of tangency. (Theorem)
Lesson Synthesis
In this lesson, students have developed the concept of distance between a point and a line and proved that lines tangent to circles are perpendicular to the radius at the point of tangency. Ask students to think about what they have learned about right angles and right triangles in this unit and in this course. Ask students to brainstorm a list, as a class, of ways to tell whether a given triangle is a right triangle and not just a triangle that looks very close to a right triangle. Here are a few possible responses:
• If the longest side of the triangle is labeled and the other two sides are labeled and , then one can check whether . If the Pythagorean Theorem doesn’t hold, then the triangle is not a right triangle. There may be a larger margin of error with this method depending on how di cult it is to measure the distances.
• Find the midpoint of the longest side. If the circle centered at that midpoint going through the two vertices of the longest side also happens to go through the third vertex, then the triangle is a right triangle, and if it doesn’t, then it’s not a right triangle. Here is an example of a triangle
where angle is not a right angle because point doesn’t lie on the circle with diameter :
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Teacher Guide