3. The Similar Triangle Method (9 points total) To prove that the Pythagorean theorem works, leave out the dimensions and represent the trunk of the car with a right triangle ABC. Drawing an altitude from vertex C to side c creates two new triangles, BCD and ACD (see diagram). a. Here are the three triangles shown separately These triangles are similar. How do you know? HINT: Compare the angle measures. (1 point) b. Complete the proportion to compare the first two triangles. (1 point) B/C=D/B D/B=F/A ? c. Cross-multiply the ratios in part b to get a simplified equation. (1 point) d. Complete the proportion to compare the first and third triangles. (1 point) e. Cross multiply the ratios in part d to get a simplified equation. (1 point) f. Complete the steps to add the equations from parts c and e. This will make one side of the Pythagorean theorem. (1 point) g. Factor out a common factor from part f. (1 point) h. Finally, replace the expression inside the parentheses with one variable and then simplify the equation to a familiar form. HINT: Look at the large triangle at the top of this problem. (2 points) Further Exploration 4. A Geometric Proof (4 points total) Here is another interesting proof. The smaller white square is inside a larger square. Each blue area is a right triangle. To find the area of the white square, you can find the area of the big square and subtract the areas of the right triangles.