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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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