geometry and measurement student pages for packet 4: … geo4-sp.pdfthe pythagorean theorem 4.1...

Name ___________________________ Period __________ Date ___________

Geometry and Measurement Unit (Student Packet) GEO3 – SP

GEO4.1 Making Sense of the Pythagorean Theorem

• Explore the Pythagorean theorem numerically, algebraically, and geometrically.

• Understand a proof of the Pythagorean theorem. • Use the Pythagorean theorem and its converse to solve

problems.

1

GEO4.2 Applications of the Pythagorean Theorem • Use the Pythagorean theorem and its converse to solve

problems. • Learn how to find perimeters and areas of triangles and

rectangles. • Learn how to find the diagonal of a rectangular prism.

9

GEO4.3 Vocabulary, Skill Builders, and Review • Review approximating square roots. • Solve equations that include squares and square roots.

14

GEO4 STUDENT PAGES

GEOMETRY AND MEASUREMENT Student Pages for Packet 4: The Pythagorean Theorem

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The Pythagorean Theorem

Geometry and Measurement Unit (Student Pages) GEO4 – SP0

WORD BANK (GEO4)

Word Definition Example or Picture

converse of the Pythagorean theorem

hypotenuse

legs of a right triangle

linear interpolation

perfect square

Pythagorean theorem

Pythagorean triple

right triangle

square of a number

square root of a number

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The Pythagorean Theorem 4.1 Making Sense of the Pythagorean Theorem

Geometry and Measurement Unit (Student Pages) GEO4 - SP1

MAKING SENSE OF THE PYTHAGOREAN THEOREM

Ready (Summary)

We will explore the relationship among the side lengths of right triangles and then look at a proof of the Pythagorean theorem. Then we will use this theorem to solve problems.

Set (Goals) • Explore the Pythagorean theorem

numerically, algebraically, and geometrically.

• Understand a proof of the Pythagorean theorem.

• Use the Pythagorean theorem and its converse to solve problems.

Go (Warmup)

Simplify each expression.

1. a + a 2. a a

3. ab+ ab 4. ab ab

5.

+1 12 2

a a

6.

2 2

a a

7.

+

1 12 2

ab ab

8.

+

2 2ab ab

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TWO RIGHT TRIANGLES

= 1 linear unit = 1 square unit

Smaller triangle Larger triangle 1. Length of the shorter leg

2. Length of the longer leg

3. Area of the square on the shorter leg

4. Area of the square on the longer leg

5. Area of the square on the hypotenuse

6. Length of the hypotenuse

7. Write a conjecture about the relationship between the area of the square on the hypotenuse and the area of the squares of the legs.

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A PROOF: PART 1

Here is a right triangle with side lengths a, b, and c:

The two large, congruent squares on the right have been made using lengths a, b, and c. 1. Label some right angles and some

lengths on several segments of both large squares.

2. Write the area of each polygonal

piece inside of it.

3. Cut out both large squares. Then cut them apart along the interior lines.

a

b c a

a

b

b

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A PROOF: PART 2 1. Write the areas inside the polygonal pieces in the two square figures above (get this

information from the pieces previously cut.) 2. Write an equation that equates the sum of the areas of the shaded polygons with the sum

of the areas of the unshaded polygons. 3. Simplify your equation. 4. Use words to state the meaning of this equation as it refers to the legs and the hypotenuse

of the original triangle. 5. This relationship is called the __________________________________________ .

b

a

c

=

b a

b

a

b a

b

a c

c c

c

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PYTHAGOREAN THEOREM PRACTICE

Pythagorean theorem: If a triangle is a right triangle, then the sum of the squares of the length of the two legs is equal to the square of the length of the hypotenuse. Converse of the Pythagorean theorem: If the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side, then the triangle is a right triangle.

Use the Pythagorean theorem and its converse to answer these questions: 1. Draw the squares on the legs and the square on the hypotenuse of the right triangle below. 2. Find the area of each square and fill in the blanks to show the Pythagorean relationship.

Area equation: (_____) + (_____) = (_____) 3. Find the length of each side of the triangle and fill in the blanks to show the Pythagorean

relationship.

Side length equation: (_____)2 + (_____)2 = (_____)

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PYTHAGOREAN THEOREM PRACTICE (continued) 4. A classmate suggests that a triangle with side lengths of 4, 5, and 9 units is a right triangle.

Use the Pythagorean theorem to show that this must be incorrect. 5. A right triangle has legs of lengths 5 and 12 units. What is the length of its hypotenuse? 6. Notice that the answer to problem #5 is a whole number. When all three sides of a right

triangle have whole number side lengths, the three numbers are called a Pythagorean triple. Are the side lengths of the triangle in problem #1 a Pythagorean triple? Explain your answer.

7. Find another example of a Pythagorean triple in this lesson. Draw the right triangle and

label its sides.

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PYTHAGOREAN THEOREM PRACTICE (continued) 8. One triangle has sides of length 4, 6 and 8 centimeters. Another triangle has sides of length

6, 8 and 10 centimeters. Is either of these triangles a right triangle? Explain. 9. A triangle has side lengths of 3, 5, and 6 units. Use the Pythagorean theorem to show why

this cannot be a right triangle. 10. Latonya said, “The third side of triangle T is 61 inches long, because

2 26 + 5 = 36 + 25 = 61 .” Is Latonya correct? Explain.

5 in.

6 in. T

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The Pythagorean Theorem 4.2 Applications of the Pythagorean Theorem


APPLICATIONS OF THE PYTHAGOREAN THEOREM

Ready (Summary)

We will solve problems that involve the use of the Pythagorean theorem or its converse.

Set (Goals) • Use the Pythagorean theorem and its

converse to solve problems. • Learn how to find perimeters and areas

of triangles and rectangles. • Learn how to find the diagonal of a

rectangular prism.

Go (Warmup)

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2, where a < c and b < c.

Determine which sets of three of numbers could be Pythagorean triples.

1. 5 12 13 2. 10 20 30

3. 15 17 8 4. 8 10 6 For

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FIND THE MISSING LENGTH 1

Find the missing length in each right triangle. If needed, use fractions to write square root approximations. Example: +

+

2 2 2

2

2

6 = 1036 = 100

= 64 = 8 units

xx

xx

1.

2.

Solve.

3. To get from home to work every day, Samos drives 7 miles east on Avenue A, and then drives north on Avenue B. He knows that the straight-line distance from his home to his place of work is about 25 miles. How many miles is his drive north on Avenue B?

a. Draw a picture:

b. Write an appropriate formula and substitute with numbers:

c. Answer the question in words:

13 ft

5 ft

v

2 cm

5 cm w

6 units

x units

10 units

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FIND THE MISSING LENGTH 2 For each problem, use fractions to write square root approximations.

1. Find the diagonal of a rectangle whose sides are 15 mm and 20 mm long.

a. Sketch and label the figure:

b. Write an appropriate formula and substitute:


2. Find the diagonal of a square whose side is 10 cm long.




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FIND THE MISSING LENGTH 2 (continued)

For each problem, use fractions to write square root approximations.

3. Find the height of an isosceles triangle with two congruent sides of 12 inches each and a base that is 18 inches long.




4. Find the height of an equilateral triangle whose sides are 4 feet in length.




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THE DIAGONAL OF A RECTANGULAR PRISM

Find the length of the diagonal of a rectangular prism with a length of 4 cm, a width of 3 cm, and a height of 12 cm.

(Prism not drawn to scale)

1. One labeled side of the rectangular prism is PY. Name two other sides of the prism. 2. The diagonal of the BASE of the prism is line segment __________. Use a colored pencil

to draw a triangle that includes this diagonal as the hypotenuse. Mark the right angle on the prism, and sketch the triangle. Find the length of the diagonal of the prism’s base.

3. The diagonal of the prism is line segment ________. Use a different colored pencil to draw

a triangle that includes this diagonal as the hypotenuse. Mark the right angle on the prism, and sketch the triangle. Find the length of the diagonal of the prism.

P

H

T

Y

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The Pythagorean Theorem 4.3 Vocabulary, Skill Builders, and Review


FOCUS ON VOCABULARY (GEO4)

Match the words to the clues. Words Clues 1. right triangle a. + =2 2 2a b c

2. hypotenuse b. The product of a number with itself.

3. legs c. 3, 4, 5 is an example of this.

4. square of a number d. The side of right triangle opposite the right angle. The longest side in a right triangle.

5. square root of a number e. If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.

6. perfect square f. A method of approximating the values of a function at points in some interval using the values at the endpoints.

7. Pythagorean theorem g. 5 is the square root of 25.

8. Pythagorean triple h. A triangle that has a right angle.

9. converse of the Pythagorean theorem i. Two sides of the triangle adjacent to the right angle.

10. linear interpolation j. A number that is a square of a natural number. Example: = 21 1 , = 24 2 , = 29 3 .

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SKILL BUILDER 1

Find the area of each figure (in square units). 1. 2.

3.

4.

5. 6.

7. 8.

9. Find the side length and area of the

square. s = _______ linear units A = _______ square units

10. Find the side length of the square

s = _______ linear units

A = 49 square units

6

x

x 6

8

x

y

6

8

x

y

6

6 x

x

6

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SKILL BUILDER 2 1. List the first 20 natural numbers and their squares.

n 1

n2 49

n 11

n2 225

Approximate each square root between two consecutive integers, as a fraction, and as a decimal rounded to the nearest tenth.

Example: 35

< <25 35 36

< <5 35 6

≈1035 511

≈35 5.9

2. 300 3. 67

4. 150 5. 125 6. 375 For Prev

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SKILL BUILDER 3

Simplify each numerical expression. 1. 62 + 82 2. 16 9+

3. 16 9+ 4. 25 4+

5. +2 23 4 6. +2 25 12

Solve each equation for the positive value of x.

7. x2 = 49 8. x + 3 = 5

9. x + 32 = 52 10. x2 + 32 = 52

11. + =2 28 10x 12. + =2 28 100x

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SKILL BUILDER 4

Determine whether each set of three numbers is a Pythagorean triple.

1. 9, 12, 15

Is 92 + 122 = 152 ? 81 + 144 = 225 ?

225 = 225 YES

2. 12, 16, 20

3. 7, 24, 25 4. 4, 6, 8

Find the missing length of each right triangle using the Pythagorean theorem. 5.

6.

7.

* Triangles are not drawn to scale.

12 mm

13 mm a

4 ft

3 ft

r

41 ft

9 ft

k

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SKILL BUILDER 5

Show all work. 1. Find the diagonal of a rectangular prism whose length is 9 in, width is 12 in, and height is

8 inches.

A gift box in the shape of a cube has a side length of 24 inches. 2. What is the volume of the cube in cubic inches? 3. What is the volume of the cube in cubic feet? (12 inches = 1 foot) 4. What is the surface area of the box in square feet?

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SKILL BUILDER 6

Show all work.

1. A cube is 5 inches on each side. Find the length of the longest stick that will fit inside the cube (that is, find the diagonal of the cube).

A cylinder has diameter is 20 mm and height is 40 mm. 2. What is the volume of the cylinder in cubic millimeters? (π ≈ 3.14) 3. What is the volume of the cylinder in cubic centimeters? (10 mm = 1 cm) 4. What is the lateral area (i.e. the surface area without the area of the bases) of the

cylinder in square centimeters?

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TEST PREPARATION (GEO4)

Show your work on a separate sheet of paper and choose the best answer. 1. Which of these numbers has a value between 11 and 12?

A. 107 B. 120 C. 136 D. 145

2. Compute 25 4+ .

A. 29 B. 7 C. 29 D. 7

3. What is the length of the missing side of the right triangle?

A. 8 cm B. 9 cm C. 10 cm D. 11 cm

4. What is the diagonal of a rectangle whose sides are 6 m and 8 m?

A. 8 m B. 9 m C. 10 m D. 11 m

5. Which sets of three numbers could represent the side lengths of a right triangle?

A. 2, 4, 6 B. 4, 6, 8 C. 6, 8, 10 D. All of these

6. A right triangle has legs of 9 cm and 12 cm. Find the length of the hypotenuse.

A. 13 cm B. 14 cm C. 15 cm D. 16 cm

15 cm 17 cm

x

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KNOWLEDGE CHECK (GEO4)

Show your work on a separate sheet of paper and write your answers on this page. 4.1 Making Sense of the Pythagorean Theorem 1. Determine whether a triangle with side lengths 4 m, 6 m, and 7 m is a right triangle. 2. A right triangle has legs of 5 mm and 12 mm. Find the length of its hypotenuse. 4.2 Applications of the Pythagorean Theorem 3. Could a right triangle have side lengths 24, 25, and 7 units? 4. A square has a perimeter of 20 cm. Find the length of its diagonal rounded to the nearest

tenth. 5. Find the height of an equilateral triangle whose side is 8 ft. Round your answer to the

nearest tenth.

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Home-School Connection (GEO4) Here are some questions to review with your young mathematician. 1. Use fractions and decimals to express the approximation for 70 . 2. Determine whether a triangle with side lengths 3 m, 4 m, and 5 m is a right triangle. 3. Find the height of an isosceles triangle with two congruent sides of 5 m and a third

side that is 6 m (use the long side as the base). Parent (or Guardian) signature ____________________________

Selected California Mathematics Content Standards

MG 7.2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the

surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders.

MG 7.3.3 Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations,

MR 7.1.2 Formulate and justify mathematical conjectures based on a general description of the mathematical question or problem posed.

FIRST PRINTING DO NOT DUPLICATE © 2009

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geometry and measurement student pages for packet 4: … geo4-sp.pdfthe pythagorean theorem 4.1...

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