1 Pythagorean Theorem

1 – Pythagorean Theorem

1. What is a right triangle? ______

______

2. Right triangles have special names for their sides. The two sides that come together forming the right angle are calledlegs. The side opposite the right angle is called the hypotenuse. The hypotenuse will ALWAYS be the longest side. (Why?) The side opposite the smallest angle will ALWAYSbe the shortest leg. (Why?)

3. The table gives the lengths of the legs and hypotenuse of various right triangles. Complete the table.

Short Leg / Long Leg / Hypotenuse / (Short Leg)² / (Long Leg)² / (SLeg)² +(LLeg)² / Hypotenuse²
3 / 4 / 5
9 / 12 / 15
6 / 8 / 10
8 / 15 / 17

a. What do you notice in the table? ______

______

b. This relationship only exists in right triangles and is called the Pythagorean Theorem.

Write in your own words a description of the Pythagorean Theorem. ______

______

4. Remember that(ShortLeg)²+ (Long Leg)² = Hypotenuse² and by algebra we also derive

(Short Leg)2 = Hypotenuse2 – (Long Leg)2 OR (Long Leg)2 = Hypotenuse2 – (Short Leg)2

Determine the length of the missing side in each triangle; round your answers to the nearest

tenthof a unit.

Show your work using words, numbers and/or diagrams(charts or graphs).

c = ______

b = ______

2- The Tennis Court

1. A tennis court measures 36 feet by 78 feet. What is the diagonal’s (a diagonal is the straight line from one vertex of a figure to an opposite vertex.) length?

Show your work using words, numbers and/or diagrams.

2. A major league baseball diamond measures 90 feet by 90 feet. Catchers are often required to throw from home plate to 2nd base. How far is the throw from home plate to 2nd base?

Show your work using words, numbers and/or diagrams.

3. An eight-meter ladder leans against a building. The base (bottom) of the ladder is three meters from the base of the building. How far is it from the top of the ladder to the base of the building?

Show your work using words, numbers and/or diagrams.

4. A 20 m pole will be held by a wire extending from the top of the pole to a stake driven into the ground 15 m away from the base of the pole. What is the length of the extended wire?

Show your work using words, numbers and/or diagrams.

3 – It’s All Square to Me

Using grid paper, cut out squareswith areas of 9, 16, 25, 36, 81, 100, 121, 144, 169 and several more of your choosing. Use three of the squares to construct a triangleas shown below. Tape the squares in positionon the colored paper. Create 5 different such triangle sets as shown, but different sizes.

Determine if the triangle formed is acute (measures between 0º and 90º) , right (measures exactly90º) or obtuse (measures between 90º and 180º)by measuring the angles. Record your data (putting the lengths of the sides of the triangle in increasing order) in the accompanying chart. Continue this process until you have measured and recorded 5 sets of squares and entered the data into your chart. In your data find a pattern similar to the Pythagorean Theorem which describes the relationship between the squares of the lengths of the sides of a triangle and the type of triangle formed.

Triangle side lengths in increasing order
a, b, c / Area of the largest square
c2 / Area of one of the smaller square
a2 / Area of the remaining square
b2 / a2 + b2 / Is the triangle formed acute, right or obtuse?
5, 6, 10 / 100 / 25 / 36 / 61 / Obtuse

b. For what type of triangle is c² = a² + b²? ______

c. If c² ≠ a² + b², for what kind of triangle is c² < a² + b²? ______

d. If c² ≠ a² + b², for what kind of triangle is c² a² + b²? ______

6. In your own words, state the “Un-Pythagorean Theorem”. ______

______

Right triangle ABC has a right angle at vertex C. Leg a = 9 inches and Leg b = 40 inches.

a. What is the length of side c, the hypotenuse? ______

Show your work using words, numbers and/or diagrams.

b. If triangle ABC from above was an acute triangle, what would you know about the length

of side c? ______

______

c. What if angle C were an obtuse angle what would you know about the length of side c?

______

4 – Fire, Fire

A surveillance (observation) helicopter reported a grass fire to the central dispatch. The dispatcher located the fire on a gridded map and determined the fire is 8 miles due east and 11 miles due north from the central dispatch (radio room). Thedispatcher (radio operator) establishes the fire at coordinates (8,11). The dispatcher determines that the fire station located at coordinates (2,3) is closest to the fire and calls that station to send out a truck. The fire truck travels in a direct route with no obstacles.

a. What are the coordinates(ordered pairs of numbers that identify points on a plane) of the right angle vertex of the triangle? ______

b. What is the horizontal (extending from side to side; parallel to the horizon)distance between the station and the fire? ______

Show your work using words, numbers and/or diagrams.

c. What is the vertical (extending straight up and down; perpendicular to the horizon)distance between the fire station and the fire? ______

Show your work using words, numbers and/or diagrams.

d. What is the actual distance from the station to the fire? ______

Show your work using words, numbers and/or diagrams.

e. A fire truck in that area travels at an average speed of 50 mph? How long will it take, in minutes, for a fire truck to get from the fire station to the fire? ______

Show your work using words, numbers and/or diagrams.

2.A surveillance helicopter reported another grass fire to the central dispatch. The dispatcher located the

fire on a gridded map and determined the fire is directly west of the central dispatch. The dispatcher determines that the fire station located at coordinates (-1, -7) is closest to the fire and calls that station to send out a truck. The fire truck travels in a direct route with no obstacles (interferences).

a. What are the coordinates of the right angle vertex (corner) of the triangle? ______

b. What is the horizontal distance between the fire station and the fire? ______

Show your work using words, numbers and/or diagrams.

c. What is the vertical distance between the station and the fire? ______

Show your work using words, numbers and/or diagrams.

d. What is the distancefrom the station to the fire? ______

Show your work using words, numbers and/or diagrams.

e. A fire truck in that area travels at an average speed of 50 mph? How long will it take, in

minutes, for a fire truck to get from the fire station to the fire? ______

Show your work using words, numbers and/or diagrams.

3. A scientist determined that twins occur in 1 out of every 80 births. The scientist uses a random

(chance)sample of 560 pregnant women.

Which represents the number of pregnant women expected to give birth to twins?

 A. 7

 B. 8

 C. 70

 D. 80

4. A commercial artist has a sketch of a rectangular logo that is 7 inches high. She needs to proportionally

(a part to a whole comparison) reproduce the logo on a sign that is 8 feet high. The sketch of the logo contains a letter M that is 5 inches tall.

Which represents how tall the letter M will be on the larger sign?

 A. 4.4 feet

 B. 5.7 feet

 C. 6.0 feet

 D. 11.2 feet

5. Damon wants to fertilize his lawn for the spring season. The dimensions (size) of his lawn are shown.

Johnson’s Garden Shop sells fertilizer in 6-pound bags that cover an area of 500 square feet.

Which represents the number of bags of fertilizer Damon will need to completely fertilize his lawn?

 A. 2

 B. 3

 C. 4

 D. 5

5– You are Similar to Me

If someone says to you that you look similar to someone else, what do they mean? Probably they mean that you look alike, almost the same. In geometry, similar figures look alike. They have the same shape, but they may not be the same size. Have you ever used a copy machine? A copy machine can enlarge or reduce the size of what is being copied. Here are some triangles that have been copied:

a. Which triangles are similar to triangle A? ______

______

b. Does it matter that some of the triangles are turned sideways or upside down? ______

c. Do they still have the same shape? ______

d. Figures that are exactly the same size and shape are called congruent. Are any of the triangles

shown congruent?______If so list the congruent triangles:______

Use these figures (shapes) to answer the following questions:

a. How are the figures alike? ______

______

b. How are the figures different? ______

______

In order for figures to be similar, one of the two conditions listed must be met:

Condition: Corresponding (or matching) angles must have the same measure.

Condition: Corresponding (or matching) sides must all be in the same ratio.

c. Are the figures all similar? ______

Show your work using words, numbers, and/or diagrams

6 – Similar Things

3. Use these figures (shapes) to answer the following questions:

a. Which are the matching or corresponding sides in the two rectangles? ______

______

b. Which are the matching or corresponding angles? ______

______

c. Are the rectangles similar? ______

(Reference (look at) both sides and angles)

Use these two triangles to answer the following questions:

If there are corresponding sides in the two triangles list them as pairs: ______

______

Looking at each corresponding pair, is there a common ratio?______

If so what is the common ratio?______

b. If there are the corresponding angles List them as pairs:______

c. Are the triangles similar? ______Explain why or why not.______

______

5. Use these shapes to answer the following questions:

If you know that two figures are similar, you can use proportions or equivalent (equal) ratios to find the lengths of the unknown sides.

a. The two rectangles shown are similar. What would be the proportion that you would use to

determine the length of the missing side? ______

b. Solve the proportion for x. ______

Show your work using words, numbers, and/or diagrams.

6. Tell whether the pair of polygons in figure 1 are similar and whether the polygons in figure 2 are similar.

a. Are the polygons in figure 1 are similar? ______Why?______

b. Are the polygons in figure 2 are similar? ______Why?______

Figure Figure

Show your work using words, numbers, and/or diagrams.

7. The right triangles shown are similar. Determine the length of the missing sides.

x = ______y = ______z = ______

n = ______p = ______

Show your work using words, numbers, and/or diagrams.

8. The water skiing ramps shown have similar triangles. What is the height of the smaller ramp?

Show your work using words, numbers, and/or diagrams.

9. A surveyor could use similar triangles to determine the distance(d) across the lake.

What is the distance across the lake? d = ______

Show your work using words, numbers, and/or diagrams.

7 – Appearance is Everything

10. Examine each of the shapes below. For each pair, decide if the two shapes appear to be similar without using any measurements. Use a ruler and protractor to make measurements to help you decide if the shapes are similar or not. Measure all lengths in centimeters units. Record your measurements on the figures and show any calculations that were performed.

a. Do the these triangles appear(look) to be similar? Yes or No

Measurements/Calculations:

Are they really similar? Yes or No? Support your conclusion. ______

______

b. Do the these triangles appearto be similar? Yes or No

Measurements/Calculations:

Are they really similar? Yes or No? Support your conclusion. ______

______

c. Do the these triangles appearto be similar? Yes or No

Measurements/Calculations:

Are they really similar? Yes or No? Support your conclusion. ______

______

d. Do these squares appearto be similar? Yes or No

Measurements/Calculations:

Are they really similar? Yes or No? Support your conclusion. ______

______

e. Do these rectangles appearto be similar? Yes or No

Measurements/Calculations:

Are they really similar? Yes or No? Support your conclusion. ______

______

f. Write a description of what it means for two shapes to be similar. ______

______

8 – Common Trusses

A truss is a rigid framework (structure) used in building bridges and roofs. Trusses are an efficient way to span long distances with a minimum of materials and still maintain strength. The following are three commonly used trusses.

Triple Howe Agricultural Truss: Depending on pitch (the ratio of vertical change to horizontal change) and spacing, these trusses can clear span up to 84 feet.

2-piece “Piggyback” trusses can achieve steep pitches over large spans.

Common attic truss can provide “Bonus Room” over garage or elsewhere.

11. The pitch of a roof is defined as the ratio of vertical change to horizontal change (also known as slope). Below is a diagram of a roof with a 2/4 pitch spanning 24 feet (segment ).

Segment = Segment (diagram is not drawn to scale)

The altitude (height) from point S is a perpendicular segment drawn to side . In isosceles triangles, the altitude bisects (cuts in half) both the angle and the side.

a. Draw the altitude from point S and label this intersection (the point where two lines cross or touch)point R.

b. Determine the length of segment ______

Show your work using words, numbers, and/or diagrams.

c. Determine the length of segment. ______

Show your work using words, numbers, and/or diagrams.

d. Determine the length of segment. Express your answer in terms of feet and inches.

Show your work using words, numbers, and/or diagrams.

e. Make a scale drawing (smaller/larger version of original drawing)of the (Triple Howe Agricultural Truss)roof on the grid. Indicate the scale factor that you chose. Be sure to label the vertices with the letters P, S and Q in the appropriate locations to match the roof. Also, be sure to label R.

Scale Factor (ratio expressing the amount of magnification): ______

f. Subdivide into four equal segments. Label each subdivision point from left to right A, B, and C. At each subdivision point draw a perpendicular to . Label the intersection of each perpendicular (at right angles to the horizon)with PS from left to right as: L, M and N. Determine the lengths of segments, and .

Show your work using words, numbers, and/or diagrams.

g. Draw segments , , and . Determine these lengths.

h. To complete the truss, reflect the figure across.

8 – Geoboard Triangles

12. Examine the three triangles on the geoboard: triangle CDB, triangle CEG and triangle CFA.

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Are the three triangles similar? ______

Why or why not? ______

______

c. Turn the geoboard so that the legs are vertical and horizontal. Find the ratio between the vertical and horizontal legs of each right triangle. ______

______

What can you say about this ratio? ______

______

13. You investigated a nest of three right triangles that shared the common acute angle at C. Create a nest (one triangle inside the other – similar to the example given) of similar triangles that are not right triangles. How do you know that the three nested triangles are similar?

______

Practice Problems

14. Lolla thought that she remembered a relationship between the lengths of the sides of a triangle and the measure of the angles across from the sides.

Which represents the relationship of the angles of from smallest to largest?

 A.

 B.

 C.

 D.

15. The club members hiked 13 kilometers north and 14 kilometers east, but then went directly home as shown by the dotted line.

Which is the distance they traveled to get home?

 A. 5.2 km

 B. 15.0 km

 C. 19.1 km

 D. 27.0 km

Student: Ch 14“Pythagorean Theorem” 6/03/08 Page 1 of 24