1
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.2
TASK 3.1.2: 30-60 RIGHT TRIANGLES
Solutions
1. Shown here is a 30-60 right triangle that has one leg of length 1 and the hypotenuse of
length 2. Call this triangle T1. 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
2
calculator answer but find your answer algebraically!) Record your
1
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 1
Hypotenuse
Process
1
2
22 ! 12
2
4
42 ! 22 = 16 ! 4 = 12
2 3
3
6
62 ! 32 = 36 ! 9 = 27
3 3
4
8
82 ! 42 = 64 ! 16 = 48
4 3
n
2n
(2n)2 ! n2 = 4n2 ! n2 = 3n2
n 3
Leg 2
3
2. Now consider the 30-60 right triangle, T2, that has one leg of length 2 and the
hypotenuse of length 4. How is this triangle related to T1? 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 2 of
4
2
triangle T2 related to the length the Leg 2 of T1?
Leg 2 of T2 is twice the length of Leg 2 of T1.
3. Repeat the process given above on T3, a 30-60 right triangle that has one leg and a
hypotenuse three times the length of the leg and hypotenuse of T1, and T4, a 30-60
right triangle that has one leg and a hypotenuse four times the length of the leg and
hypotenuse of T1. Record the results in Table I. How are the lengths of the
hypotenuse of T3 and T4 related to the length the hypotenuse of T1?
Leg 2 of T3 and Leg 2 of T4 are three and four times the length of Leg 2 of T1,
respectively.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
2
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.2
4. Suppose you are given a 30-60 right triangle with one leg of length n and hypotenuse
of length 2n. 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 2 of this triangle will be n times as long as the
second leg of T1.
5. Suppose you are given a 30-60 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 = (2n)2 ! n2 = 3n2 . So by squaring both sides of this equation, we see that
3n2 = 1296 and then by dividing both sides by 3, n2 = 432 . Thus by taking square
roots of both sides, we see that the short leg of the triangle must have length of
432
(approximately 20.78 units). The hypotenuse then has length of 2 432 units
(approximately 41.57 units).
6. Calculate the area of each of the triangles from Table I and record your results in
Table II.
Table II
Triangle
Leg 1
Leg 2
T1
1
3
T2
2
2 3
T3
3
3 3
T4
4
4 3
Tn
n
n 3
Process
1
* (1) *
2
( 3)
1
* (2) * ( 2 3 )
2
1
* (3) * ( 3 3 )
2
1
* (4) * ( 4 3 )
2
1
* (n) * ( n 3 )
2
Area
3
2
4 3
2
9 3
2
16 3
2
n2 3
2
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
3
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.2
7. Using your table and your rule, determine the exact area for a 30-60 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 30-60 right triangle with short leg
62 3
= 18 3 square units.
2
When you use the trace feature on your
calculator, you will see that when n=6, the
area is 31.176915. These values are not the
same! 31.176915 is an approximation of
of length 6 is
18 3 , the exact answer.
8. Describe the dimensions of T2 and T4. How is the area of triangle T2 related to the
area of triangle T4?
The dimensions of T4 are twice the dimensions of T2 and the area of T4 is four times that
of T2.
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
64 3
.
2
10. How is the area of T2 related to the area of T8? Explain this relationship for T2 and
T8 based on information gained from questions # 8 and #9.
The dimensions of T8 are 4 times the dimensions of T2 and the area of T8 is sixteen times
that of T2. 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 T2. So the
dimensions of T8 are 4 times those of T2. Similarly the area of T8 is four times that
of T4 which is four times that of T2. Thus the area of T8 is 16 times the area of T2.
11. Is triangle T4 a dilation of triangle T2? If not, why not? If so, why and are there
other 30-60 right triangles that are dilations of T2? Explain.
Yes. T4 is a dilation of T2 with a scale factor of 2. There are lots of others that are also
dilations of T2. T8 is a dilation of T2 with a scale factor of 4. T3 is a dilation of T2
with a scale factor of 3/2. T1 is a dilation of T2 with a scale factor of 1/2.
Participants should be expected to see why each of these hold true.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
4
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.2
TASK 3.1.2: 30-60 RIGHT TRIANGLES
1. Shown here is a 30-60 right triangle that has one leg of length 1 and the
hypotenuse of length 2. Call this triangle T1. 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
2
algebraically!) Record your answer and the process you
1
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 1
Hypotenuse
1
2
Process
Leg 2
2. Now consider the 30-60 right triangle, T2, that has one leg of length 2 and the
hypotenuse of length 4. How is this triangle
related to T1? Calculate the exact length of the
4
second leg. Record your answer and the process
2
you used in Table I. How is the length of the
hypotenuse of triangle T2 related to the length the
hypotenuse of T1?
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
5
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.2
3. Repeat the process given above on T3, a 30-60 right triangle that has one leg and
a hypotenuse three times the length of the leg and hypotenuse of T1, and T4, a 3060 right triangle that has one leg and a hypotenuse four times the length of the leg
and hypotenuse of T1. Record the results in Table I. How are the lengths of the
hypotenuse of T3 and T4 related to the length the hypotenuse of T1?
4. Suppose you are given a 30-60 right triangle with one leg of length n and
hypotenuse of length 2n. 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 30-60 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 1
T1
1
T2
2
T3
3
T4
4
Tn
n
Leg 2
3
Process
1
* (1) *
2
( 3)
Area
3
2
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
6
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.2
7. Using your table and your rule, determine the exact area for a 30-60 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 T2 and T4. How is the area of triangle T2
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.
10. How is the area of T2 related to the area of T8? Explain this relationship for T2
and T8 based on information gained from questions (8) and (9).
11. Is triangle T4 a dilation of triangle T2? If not, why not? If so, why and are there
other 30-60 right triangles that are dilations of T2? Explain.
November 23, 2004. Ensuring Teacher Quality: Algebra I, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.