Shape 

Copyright David & Cynthia Thomas, 2009

 

Sides, Angles, Interior Lines & Points--Exercises

 

Part I: Do the directed activities in this section.

 

Part II: Complete the following exercises

1)      Using the Pythagorean Theorem, find the missing segment in the following triangles:

Side

Side

Hypotenuse

1cm

1cm

 

1cm

 

2cm

 

36cm

39cm

 

2)      Find Pythagorean Triples based on the following integers {m,n}

a)      {7,6}

b)      {5,2}

c)      {3,1}

3)      Use Excel to prove that all sets of the form {m2 – n2, 2mn, m2 + n2} satisfy the Pythagorean relationship.

4)      Use The Geometers Sketchpad and model 2_Circumcenter.gsp to reposition points A, B, and C. 

a)      Under what conditions does the circumcenter lie inside the triangle?  

b)      Under what conditions does the circumcenter lie on the triangle? 

c)      Under what conditions does the circumcenter lie outside the triangle?  

5)      What logical argument can you offer justifying the observation that in every triangle the circumcenter is equidistant from the vertices?

6)      Use The Geometers Sketchpad and model 2_Incenter.gsp to find the shortest distance from the incenter to each side of the triangle.  Along what interior lines would you measure this distance?  Where would those lines intersect the sides of the triangle?

7)      Use The Geometers Sketchpad and model 2_Centroid.gsp to measure the area of the shaded portion of ΔABC.  Construct and measure the areas of the other triangular regions indicated in Figure 2.20.  What relationship exists between these area measurements? 

8)      Use the following models to prove the Pythagorean Theorem. 

            Pythagorean Model 1

            Pythagorean Model 2

The four colored triangles are congruent

with hypotenuse c and sides a and b.

  The four colored triangles are congruent

   with hypotenuse c and sides a and b.

9)      You've just picked up a ground ball at first base, and you see the other team's player running towards third base. How far do you have to throw the ball to get it from first base to third base, and throw the runner out? Bases are all 90′ apart.

10)  You're locked out of your house and the only open window is on the second floor, 25 feet above the ground. You need to borrow a ladder from one of your neighbors. There's a bush along the edge of the house, so you'll have to place the ladder 10 feet from the house. What length of ladder do you need to reach the window?

11)  Create a Concept Card about the Pythagorean Theorem

12)  Create a Concept Card about Pythagorean Triples 

13)  Create a Concept Card about the circumcenter

14)  Create a Concept Card about the incenter

15)  Create a Concept Card about the centroid