September 9

Introduction to Sketchpad: Incenter and Circumcenter of Triangles

Angle Bisector The original angle to be bisected is <ABC which is seen below in black.

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Step 1: Draw an angle using the line segment tool and label it ABC using the label tool. Step 2: Select one side and and then select Construct -> Point on Object to put a point D on that side. Step 3: Select the two points A and D in that order holding the Shift key Step 4: Then select Construct -> Circle By Center+Point to construct a circle with A as the center and D as to the point. (the red circle) Step 5: Construct the point of intersection E of circle AD and side AC of <ABC by selecting Construct -> Point at Intersection Step 6: Construct circles ED and DE ( the blue circles) Step 7. Construct the points of intersection of the two circles. Step 8: Construct the line segment between A and F by selecting the two points A and F and then Construct -> Segment Step 9: Hide all of the circles by selecting them and then select Display -> Hide Step 10: Measure the angles: <CAF and <FAB by holding the shift key down and selecting the three points which define the angle. Then select Measure -> Angle. Repeat this process to obtain the measure of the other angle.

Did you create the angle bisector? Drag either C or B around and see what happens to the measure of the two angles.

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Distance The distance between two points is easy. Just connect the two points and measure the distance. However, what is the shortest distance between a point P and a line ?

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Step 1: Place a point A and a line not passing through point A. Step 2: Select the line and and then select Construct -> Point on Object to put a point D on that line. Step 3: Holding the shift key, select both points A and D and then select Construct -> Segment. Step 4: Select the segment and then Measure -> Length. Step 5: Select the point A and the line and then Measure -> Distance. Step 6: Drag point D until the length AD = the distance from A to . Step 7: Holding the shift key down, select point A and line and then select Construct -> Perpendicular.

What relationship do you see between between the shortest distance and the perpendicular?

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Incenter The point of concurrency of the three angle bisectors is called the incenter.

Step 1: Construct a triangle ABC Step 2: Construct the bisectors of < A and < C and their point on intersection, D Step 3: Construct the bisector of < B and see if it passes through point D. Step 4: Measure the distances from D to each of the three sides. What conclusion do you have about the distances from the incenter to the respective sides of the triangle? How would you draw a circle inscribed in your triangle? Step 5: Construct your circle inscribed in your triangle.

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Circumcenter The point of concurrency of the three perpendicular bisectors of the sides of a triangle is called the circumcenter.

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Step 1: Construct a triangle ABC Step 2: Construct the midpoints, E. F, and D of the three sides by holding the shift key down and selecting the three sides and then Construct -> Point at Midpoint Step 3: Construct lines perpendicular to each side through its respective midpoint, by holding the shift key down and selecting the midpoint and its side and then selecting: Construct -> Perpendicular Step 4: Construct the point of intersection, G, of these lines. Step 5: Construct a circle with center G and radius endpoint A. Does the circle you construct go through all three vertices? Notice that as you move a vertex, the point, G, may fall inside, on, or outside the triangle. Step 6: Measure < ABC and drag B so that the point G falls within the triangle, one a side of the triangle and outside the triangle/

What relationship exists between the measure of <ABC and where point G lies?

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