1
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.1
TASK 3.1.1: 45-45 RIGHT TRIANGLES
Solutions
1. On the grid below is a 45-45 right triangle with legs of length 1. Call this triangle
T1. Calculate the exact length of the hypotenuse of the triangle. (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 hypotenuse of this triangle? What special triangle is this?
Using Pythagorean’s Theorem, we see that the length of the hypotenuse is
triangle is an isosceles triangle since two sides are the same length.
2 . This
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
3.1.1
Table I
Length of
Legs
Process
Length of Hypotenuse
1
12 + 12 = 2
2
2
22 + 22 = 8
8=2 2
3
32 + 32 = 18
18 = 3 2
4
42 + 42 = 32
32 = 4 2
n
n2 + n2 = 2n2
2n2 = n 2
2. Create a 45-45 right triangle with legs of length 2 by extending
the base leg of T1 and then going up 2 units. Call this triangle
T2. Calculate the exact length of the hypotenuse of T2. Record
this information in Table I. How is the hypotenuse of T2
related to T1?
The length of the hypotenuse of T2 is twice the length of the
hypotenuse of T1.
3. Create a 45-45 right triangle with legs of length 3 by extending
the base leg of T1 and then going up 3 units. Call this triangle
T3. Calculate the exact length of the hypotenuse of T3. Record
this information in Table I. How is the hypotenuse of T3
related to T1?
The length of the hypotenuse of T3 is three times the length of the
hypotenuse of T1.
4. Create a 45-45 right triangle with legs of length 4 by extending
the base leg of T1 and then going up 4 units. Call this triangle
T4. Calculate the exact length of the hypotenuse of T4.
Record this information in Table 1. How is the hypotenuse of
T4 related to T1?
The length of the hypotenuse of T4 is four times the length of 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.
3
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.1
5. Suppose you are asked to find the length of the hypotenuse of a 45-45 right
triangle with legs of length n. Make a prediction of the length of the hypotenuse
and explain the reasoning behind your prediction.
Answers will vary. Algebraically, participants could use n in the Pythagorean
formula as the length for the legs of the triangle to get that the length of the
hypotenuse is n 2 . Participants can also observe this in the table. Geometrically,
one can see the with each unit increase in the length of the legs, the hypotenuse
increases by
length 1.
2 , the length of the hypotenuse of a 45-45 right triangle with legs of
6. Using your graphing calculator, create a scatter plot of the
data in Table I. Graph your rule for this data. Describe
your graph.
The rule produces a line through our data points.
7. Describe how triangle T2 is related to triangle T1. What is
the geometric term used to describe this relationship?
Each leg of T2 is twice the length of the corresponding leg of T1. This represents a
dilation. Participants may also say that each component of T1 was changed by a
scale factor of 2 to get triangle T2.
8. Note that T1 had legs of length 1 unit and an area of 1/2 square units as shown in
Table II. Complete Table II. If a 45-45 right triangle has an area of 32 square
units, what is the length of the of the triangle’s legs? Explain the process you
used to determine your answer.
n2
Algebraically,
= 32 . This means n2 = 64 and n=8. Thus a 45-45 right with legs
2
8 units long will have an area of 32 square units.
Table II
Length of
Legs
1
2
3
4
n
Process
Area of Triangle
1
1
* b * h = * (1) * (1)
2
2
1
*2*2
2
1
* 3* 3
2
1
*4*4
2
1
*n*n
2
1
2
2
9
2
8
n2
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.
4
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.1
Math notes
The instructor is encouraged to begin this task by discussing the difference between exact
answers and the estimated answers often obtained when using the calculator.
The topic of simplifying radicals has become a touchy subject for educators. There is
much discussion over whether or not it is necessary for students (specifically high school
students) to be taught to simplify radicals. This task works well regardless of the
perspective participants might have. The pattern is revealed very nicely geometrically. A
wonderful proof that
triangles T1 and T2.
8 = 2 2 is clear if one examines the relationship between the
In problem 7, triangle T2 is a dilation of triangle T1 with a scale factor of 2. Triangle T4
is a dilation of triangle T2 with a scale factor of 2 as well. Moreover triangle T4 is a
dilation of triangle T1 with a scale factor of 4. Also T3 is a dilation of T1 with a scale
factor is 3.
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
3.1.1
TASK 3.1.1: 45-45 RIGHT TRIANGLES
1. On the grid below is a 45-45 right triangle with legs of length 1. Call this triangle
T1. Calculate the exact length of the hypotenuse of the triangle. (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 hypotenuse of this triangle? What special triangle is this?
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
3.1.1
Table I
Length of
Legs
Process
Length of Hypotenuse
1
2
3
4
n
2. Create a 45-45 right triangle with legs of length 2 by extending the base leg of T1
and then going up 2 units. Call this triangle T2. Calculate the exact length of the
hypotenuse of T2. Record this information in Table I. How is the hypotenuse of
T2 related to T1?
3. Create a 45-45 right triangle with legs of length 3 by extending the base leg of T1
and then going up 3 units. Call this triangle T3. Calculate the exact length of the
hypotenuse of T3. Record this information in Table 1. How is the hypotenuse of
T3 related to T1
4. Create a 45-45 right triangle with legs of length 4 by extending the base leg of T1
and then going up 4 units. Call this triangle T4. Calculate the exact length of the
hypotenuse of T4. Record this information in Table I. How is the hypotenuse of
T4 related to T1?
5. Suppose you are asked to find the length of the hypotenuse of a 45-45 right
triangle with legs of length n. Make a prediction of the length of the hypotenuse
and explain the reasoning behind your prediction.
6. Using your graphing calculator, create a scatter plot of the data in Table I. Graph
your rule for this data. Describe your graph.
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.
7
Algebra I: Strand 3. Quadratic and Nonlinear Functions; Topic 1. Pythagorean Theorem; Task
3.1.1
7. Describe how triangle T2 is related to triangle T1. What is the geometric term
used to describe this relationship?
8. Note that T1 had legs of length 1 unit and an area of 1/2 square units as shown in
Table II. Complete Table II. If a 45-45 right triangle has an area of 32 square
units, what is the length of the of the triangle’s legs? Explain the process you
used to determine your answer.
Table II
Length of
Legs
1
Process
Area of Triangle
1
1
* b * h = * (1) * (1)
2
2
1
2
2
3
4
n
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.