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GEOMETRY CHAPTER 5
RELATIONSHIPS WITHIN TRIANGLES
Unit 1 Midsegment Theorem and Coordinate Proof
1. Place the figure in a coordinate plane in a convenient way. Assign coordinates to each vertex.
a. Right triangle b. Isosceles right triangle
leg lengths 2 units and 3 units leg lengths 7 units
2. Use [pic]GHJ, where D, E, and F are midpoints of the sides.
a. If DE = 4x + 5 and GJ = 3x + 25, what is DE?
b. If HJ = 8x – 2 and DF = 2x + 11, what is HJ?
3. The points T(2, 1), U(4, 5), and V(7, 4) are the midpoints of the sides of a triangle. Graph the three midsegments and use the graph to draw the original triangle. Give the coordinates of each vertex.
Unit 2 Use Perpendicular Bisectors
1. Find the length of [pic].
a. b.
2. Make up a problem similar to #1 and solve your problem.
3. Complete the statement with always, sometimes, or never. Justify your answer with a sketch or explanation.
a. The circumcenter of a scalene triangle is ________ inside the triangle.
b. If the perpendicular bisector of one side of a triangle goes through the opposite vertex, then the triangle is ________ isosceles.
c. The perpendicular bisectors of a triangle intersect at a point that is ________ equidistant from the midpoints of the sides of the triangle.
4. A cell phone tower is to be built to serve the cities of Oregon, Stoughton, and McFarland. Draw a sketch and explain how to locate the tower so it is equidistant from each city.
Unit 3 Use Angle Bisectors of Triangles
1. Find the value of x.
a. b.
2. Find the value of x that makes N the incenter of the triangle.
a. b.
3. Make a sketch and prove the angle bisector theorem.
Unit 4 Use Medians and Altitudes
1. Name the four types of points of concurrency introduced in this chapter. Explain what is true about each of these points.
2. G is the centroid of triangle ABC, AD = 8, AG = 10, and CD = 18. Find each of the following distances.
a. BD
b. AB
c. EG
d. AE
e. DG
f. CG
3. Find the coordinates of the centroid P of triangle ABC.
a. A(-1, 2), B(5, 6), C(5, -2)
b. A(0, 4), B(3, 10), C(6, -2)
4. Is [pic] a perpendicular bisector, angle bisector, median, or altitude of triangle ABC? There may be more than one correct answer.
a. b.
Unit 5 Use Inequalities in a Triangle
1. Explain how you can tell from the angle measures of a triangle which side is the longest and which side is the shortest.
2. List the sides and the angles from smallest to largest.
3. Is it possible to construct a triangle with side lengths 6, 7, and 11? Explain.
4. If two sides of a triangle measure 5 inches and 12 inches, what are the possible lengths of the third side of the triangle?
5. Explain why the hypotenuse of a right triangle must always be longer than either leg.
6. You are given a 24 cm loop of string. You want to form a triangle out of the string so that the length of each side is a whole number. Sketch and label the lengths of the sides of each of the following triangles.
a. Equilateral
b. Right
c. Isosceles
d. Scalene
Unit 6 Inequalities in Two Triangles and Indirect Proof
1. Complete with , or =.
a. DE _____ EF b. JK _____ LM
c. m(1 _____ m(2 d. m(1 _____ m(2
2. Explain why the student’s reasoning is not correct.
3. Write an indirect proof.
Given: x + y ( 14 and y = 5
Prove: x ( 9
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