Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task 3.1.2
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1 Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic. Pythagorean Theorem; Task 3.. TASK 3..: RIGHT TRIANGLES Solutions. Shown here is a right triangle that has one leg of length and the hypotenuse of length. Call this triangle T. Calculate the exact length of the second leg. (Note that the answer that you get from the calculator on this problem is not the exact answer but is an approximation of the answer. Do not use your calculator answer but find your answer algebraically!) Record your answer and the process you used in Table I. What is the name of the process that allows you to calculate the length of the other leg of this triangle? Pythagorean s Theorem is the process participants will use to find the length of the second leg of the triangle. Table I Leg Hypotenuse Process Leg! 3 4 4! = 6! 4 = ! 3 = 36! 9 = ! 4 = 64! 6 = n n (n)! n = 4n! n = 3n n 3. Now consider the right triangle, T, that has one leg of length and the hypotenuse of length 4. How is this triangle related to T? Calculate the exact length of the second leg. Record your answer and the process you used in Table I. How is the length of Leg of 4 triangle T related to the length the Leg of T? Leg of T is twice the length of Leg of T. 3. Repeat the process given above on T3, a right triangle that has one leg and a hypotenuse three times the length of the leg and hypotenuse of T, and T4, a right triangle that has one leg and a hypotenuse four times the length of the leg and hypotenuse of T. Record the results in Table I. How are the lengths of the hypotenuse of T3 and T4 related to the length the hypotenuse of T? Leg of T3 and Leg of T4 are three and four times the length of Leg of T, respectively. November 3, 004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
2 Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic. Pythagorean Theorem; Task Suppose you are given a right triangle with one leg of length n and hypotenuse of length n. Using your table, predict the length of the second leg of the triangle. Support your prediction with an algebraic argument. Participants should predict that Leg of this triangle will be n times as long as the second leg of T. 5. Suppose you are given a right triangle with the long leg having length 36. Determine the length of the short leg and of the hypotenuse. Explain how you determined each of these dimensions. 36 is the length of the second leg. Using Pythagorean s Theorem, 36 = (n)! n = 3n. So by squaring both sides of this equation, we see that 3n = 96 and then by dividing both sides by 3, n = 43. Thus by taking square roots of both sides, we see that the short leg of the triangle must have length of 43 (approximately 0.78 units). The hypotenuse then has length of 43 units (approximately 4.57 units). 6. Calculate the area of each of the triangles from Table I and record your results in Table II. Table II Triangle Leg Leg Process Area T 3 T 3 T T Tn n n 3 *() * 3 * () * 3 *(3) * 3 3 * (4) * 4 3 * (n) * n n 3 November 3, 004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
3 3 Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic. Pythagorean Theorem; Task Using your table and your rule, determine the exact area for a right triangle with short leg of length 6. Enter the area function in your calculator and graph; then use the trace feature to find the area for the specific triangle. Are the two values the same? Explain. The area of a right triangle with short leg of length 6 is 6 3 = 8 3 square units. When you use the trace feature on your calculator, you will see that when n=6, the area is These values are not the same! is an approximation of 8 3, the exact answer. 8. Describe the dimensions of T and T4. How is the area of triangle T related to the area of triangle T4? The dimensions of T4 are twice the dimensions of T and the area of T4 is four times that of T. 9. Describe the dimensions of T4 and T8. Predict the area of T8. Explain how the area of T8 relates to T4. The dimensions of T8 are twice the dimensions of T4 and the area of T8 is four times that of T4. Thus the area of T8 is How is the area of T related to the area of T8? Explain this relationship for T and T8 based on information gained from questions # 8 and #9. The dimensions of T8 are 4 times the dimensions of T and the area of T8 is sixteen times that of T. This can be seen by combining the results from (8) and (9). The dimensions of T8 are twice those of T4 which are twice those of T. So the dimensions of T8 are 4 times those of T. Similarly the area of T8 is four times that of T4 which is four times that of T. Thus the area of T8 is 6 times the area of T.. Is triangle T4 a dilation of triangle T? If not, why not? If so, why and are there other right triangles that are dilations of T? Explain. Yes. T4 is a dilation of T with a scale factor of. There are lots of others that are also dilations of T. T8 is a dilation of T with a scale factor of 4. T3 is a dilation of T with a scale factor of 3/. T is a dilation of T with a scale factor of /. Participants should be expected to see why each of these hold true. November 3, 004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
4 Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic. Pythagorean Theorem; Task TASK 3..: RIGHT TRIANGLES. Shown here is a right triangle that has one leg of length and the hypotenuse of length. Call this triangle T. Calculate the exact length of the second leg. (Note that the answer that you get from the calculator on this problem is not the exact answer but is an approximation of the answer. Do not use your calculator answer but find the answer algebraically!) Record your answer and the process you used in Table I. What is the name of the process that allows you to calculate the length of the other leg of this triangle? Table I Leg Hypotenuse Process Leg. Now consider the right triangle, T, that has one leg of length and the hypotenuse of length 4. How is this triangle related to T? Calculate the exact length of the 4 second leg. Record your answer and the process you used in Table I. How is the length of the hypotenuse of triangle T related to the length the hypotenuse of T? November 3, 004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
5 5 Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic. Pythagorean Theorem; Task Repeat the process given above on T3, a right triangle that has one leg and a hypotenuse three times the length of the leg and hypotenuse of T, and T4, a right triangle that has one leg and a hypotenuse four times the length of the leg and hypotenuse of T. Record the results in Table I. How are the lengths of the hypotenuse of T3 and T4 related to the length the hypotenuse of T? 4. Suppose you are given a right triangle with one leg of length n and hypotenuse of length n. Using your table, predict the length of the second leg of the triangle. Support your prediction with an algebraic argument. 5. Suppose you are given a right triangle with the long leg having length 38. Determine the length of the short leg and of the hypotenuse. Explain how you determined each of these dimensions. 6. Calculate the area of each of the triangles from Table I and record your results in Table II. Table II Triangle Leg Leg T 3 T T3 3 T4 4 Process *() * ( 3 ) Area 3 Tn n November 3, 004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
6 6 Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic. Pythagorean Theorem; Task Using your table and your rule, determine the exact area for a right triangle with short leg of length 6. Enter the area function in your calculator and graph, then use the trace feature to find the area for the specific triangle. Are the two values the same? Explain. 8. Describe the linear dimensions of T and T4. How is the area of triangle T related to the area of triangle T4? 9. Describe the linear dimensions of T4 and T8. Predict the area of T8. Explain how the area of T8 relates to T4. 0. How is the area of T related to the area of T8? Explain this relationship for T and T8 based on information gained from questions (8) and (9).. Is triangle T4 a dilation of triangle T? If not, why not? If so, why and are there other right triangles that are dilations of T? Explain. November 3, 004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
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