SIMILAR TRIANGLES AND TRIGONOMETRY

 EXPLORATION * TITLE
 13-1 THE SSS SIMILARITY THEOREM
13-2 THE AA AND SAS SIMILARITY THEOREMS

 13-2P

 TRIANGLES IN A TRAPEZOID
13-3 THE SIDE SPLITTING THEOREM

 

EXPLORATION 13-1. THE SSS SIMILARITY THEOREM

Objective: Show that three sets of equal ratios will produce similar triangles

1. CREATE A NEW DOCUMENT with the variable simsss.

2. Create overlapping triangles CAT and FAR with common vertex A and segment FR parallel to segment CT, as shown to the right. You will have to construct a parallel line FR, then make an overlapping segment FR. Hide the parallel line (but not the segment) that is necessary in the construction.

Drag around the vertices of triangle CAT to ensure that FR remains parallel to CT.

3. Measure and label the lengths of all of the sides of both triangles.

4. Use the Calculate tool to compute the following ratios:
CA , CT , and TA .
FA FR RA

Drag around point F or point R and observe the changes in your calculations. What can you conclude about these ratios?

 

 

 

5. Without measuring, explain below why the corresponding angles in CAT are equal to those in triangle FAR.

 

 

 

 

 

6. Given that three corresponding ratios are equal, is this enough to verify that triangle FAR ~ triangle CAT? Explain why or why not below.

 

 

 

 

EXPLORATION 13-2. THE AA AND SAS SIMILARITY THEOREMS

Objective: Show that the AA and SAS conditions produce similar triangles

1. CREATE A NEW DOCUMENT with the variable simsas.

2. Construct triangles ABC and EBD by creating two parallel lines and two intersecting segments between them, as shown to the right.

3. Create overlapping segments AC and DE. Hide the two lines.

4. Without measuring, state below three pairs of congruent corresponding angles. Explain why the pairs of angles are congruent.

 

 

 

5. Measure and label the lengths of the sides in both triangles. Use the Calculation tool to compute
 AB, AC , and CD
 BE DE BD

6. Drag the vertices of the triangles around and observe the changes in ratios. Given that three corresponding angles are congruent (AAA), is this enough to verify that triangle ABC ~ triangle EBD? Explain why or why not below.

 

 

 

7. Is it possible to show triangle ABC ~ triangle EBD with only two pairs of congruent corresponding angles (AA)? Explain why or why not below.

 

 

 

 

8. Clear the screen. Construct two dotted concentric circles (same center) with center A. Create two non-overlapping isosceles triangles ABC and ADE, whose legs are equal to the radii of their respective circles, as shown to the right.

9. Measure and label < DAE and < CAB. Drag point E around the circle until both angles have the same measurement.

10. Without measuring, explain below why
 AD = AE
AC AB

11. Measure and label DE, CB, <EDA, <DEA, <ACE, and <ABC.

 Use the Calculate tool to compute  
 DE
 CB

12. Modify the larger circle's size and observe the changes in corresponding ratios and angle measurements. Given that two corresponding ratios are equal and an included pair of angle is congruent (SAS), is this enough to verify that triangle ABC ~ triangle ADE? Explain why or why not below.

 

 

 

 

EXPLORATION 13-2P. TRIANGLES IN A TRAPEZOID

Objective: Find the relationship between the area of triangles in a trapezoid.

1. CREATE A NEW DOCUMENT with the variable tritrap.

 2. Construct a trapezoid, diagonals, and intersection of diagonals as shown to the right.

3. Which triangles in the diagram are similar? Explain how you know.

 

 

 

4. Find an equation relating the ratio of the bases of the trapezoid, the area of triangle CED and the area of triangle AEB. Explain how you determined this result.

 

 

 

5. Which triangles in the diagram have the same area? Explain why this is true.

 

 

 

 

EXPLORATION 13-3. THE SIDE SPLITTING THEOREM

Objective: Verify the side-splitting theorem and its converse

1. OPEN the file simsss in your folder.

2. Delete all of the measurements and calculations, except for the lengths of FA and RA, as shown to the right.

3. Measure the lengths of segments FC and RT. Use the Calculate tool to compute the ratios
 AF and AR
FC RT

4. Drag around point F or point R and observe changes in the calculated ratios.

5. If a parallel segment (AG) splits two sides of a triangle, what can you conclude about the ratios of the split sides? Explain below.

 

 

 

 

6. CREATE A NEW DOCUMENT with the variable sidespl.

7. Create overlapping triangles LUH and AUG, as shown to the right. Segment AG should not be drawn parallel to segment LH. Create an overlapping segment LH.

8. Measure and label the slopes of segments AG and LH, and the lengths of segments UA, AL, UG, and GH.

9. Use the Calculate tool and compute the ratios
 AL and GH
UA UG

 10. Drag point A or point G until
 AL = GH
UA UG

Observe the changes in the slope of AG and LH. If a segment (AG) splits two sides of a triangle into equal proportions, what can you conclude about that segment and the base of the triangle (LH)? Explain why below.

 

 

 

 


 *Note: These investigations cover only the first three sections of chapter 13. It is possible to create explorations for the remaining sections.

 

These activities have been designed by Evan Glazer at the University of Georgia, and Phil Gartner at Glenbrook South High School in Glenview, IL. These resources should only be used for nonprofit purposes. Contact eglazer@coe.uga.edu or pgartner@glenbrook.k12.il.us if you have questions or comments.

Last revised: August 2000