Side Splitting Theorem Investigation

Follow the instructions below to investigate the Side Splitting Theorem. Can you develop a proof of the theorem?
1. Recall what you learned in Coordinate Algebra last year. What is the slope-intercept form of a line? 2. Given the slope-intercept form of two different lines, how can you determine if they are parallel? 3. Click the topmost “Show” checkbox. Move the b_((AB)) slider. What changes as you move the slider? What stays the same? What do b_((AB)) and m_((AB)) represent? (Refer to your answer to question 1.) 4. What is the relationship between the line and side (AB) ̅? 5. How many triangles are formed when the line intersects triangle ∆ABC? 6. Click the topmost Show box until the line disappears. Click the second Show box. Slide the b_((AC)) slider. What differences do you notice between the new line and the line y=m_((AB)) x+ b_((AB))? Do the lines have anything in common? (Don’t say that they’re both lines.) 7. Click the second Show box until the line disappears. Click the third Show box. There are now segment lengths displayed. Move the b_((BC)) slider (a lot!). What is the relationship between the line and side (BC) ̅? 8. What does this line have in common with the first two lines you looked at? Is this answer the same as what you said for question 7? 9. Continue to move the b_((BC)) slider. What happens to the lengths of the segments? 10. Can you make a conjecture about some relationship between the various segment lengths? 11. Click “Show split side ratios.” What do you think “split side” means? What do the various ratios represent? 12. Move the slider b_((BC)). Describe the relationships between the ratios. Can you edit your conjecture from question 11 or make a new conjecture about the relationships between the various segment lengths? 13. Click the “Show split side ratios” box until the ratios disappear. Click the third Show box so that the line disappears and re-click the first Show box. What ratios do you think are equal based on question 12? Write as many equations as you can (that you think are true). 14. Repeat question 13 with the second line y=m_((AC)) x+b_((AC)). 15. Click the Show boxes until only the third and last Show boxes are checked. Name the two triangles that are having their side lengths measured. 16. What is the relationship between the ratios of the side lengths of the triangles? Does the relationship hold when you move the slider? 17. Recall from earlier in Unit 1. What name is given to figures that have proportional side lengths? 18. Recall the four ways of proving that triangles are similar. Prove that ∆AGF ~ ∆ABC without using the side lengths. 19. What does your result from question 16 mean for the second part of question 14?