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7th Grade

Drawing Geometric Figures

2017­02­28 www.njctl.org

Table of Contents

Determining if a Triangle is Possible Geometric Constructions: The Basics Videos Using Geometer's Sketchpad Video: Constructing Circles Video: Constructing Isosceles Triangles Video: Constructing Equilateral Triangles Video: Congruent Triangles Teacher Notes Constructions are shown within the lesson using paper and

• pencil. There are also video links

showing how to do the constructions using Geometer's

• Sketchpad. A free version of this

program is available at: http://info.mheducation.com/ sketchpad.trial.html Glossary

Click on a topic to go to that section

Determining if a Triangle is Possible

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• f Contents

How many different acute triangles can you draw? Recall that triangles can be classified according to their side lengths and the measure of their angles. Sides: Scalene ­ no sides are congruent Isosceles ­ two sides are congruent Equilateral ­ all three sides are congruent Angles: Acute ­ all three angles are acute Right ­ contains one right angle Obtuse ­ contains one obtuse angle

Drawing Triangles

How many different right scalene triangles can you draw? Recall that triangles can be classified according to their side lengths and the measure of their angles. Sides: Scalene ­ no sides are congruent Isosceles ­ two sides are congruent Equilateral ­ all three sides are congruent Angles: Acute ­ all three angles are acute Right ­ contains one right angle Obtuse ­ contains one obtuse angle

Drawing Triangles

There is another property that applies to triangles: Click the lab below to learn about the Triangle Inequality Property. Triangle Inequality Lab

Triangle Inequality Property

Triangle Inequality: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. What does this mean? If you take the three sides of a triangle and add them in pairs, the sum is greater than (not equal to) the third side. If that is not true, then it is not possible to construct a triangle with the given slide lengths.

Triangle Inequality Property

Example

Determine if sides of length 5 cm, 8 cm and 12 cm can form a triangle? Test all three pairs to see if the sum is greater: 5 + 8 > 12 5 + 12 > 8 8 + 12 > 5 13 > 12 17 > 8 20 > 5 Yes, it is possible to construct a triangle with sides of lengths 5 cm, 8 cm and 12 cm.

Example

Determine if sides of length 3 ft, 4 ft and 9 ft can form a triangle? Test all three pairs to see if the sum is greater: 3 + 4 > 9 3 + 9 > 4 4 + 9 > 3 7 > 9 12 > 4 13 > 3 No, it is not possible to construct a triangle with sides of lengths 3 ft, 4 ft and 9 ft.

Determine if triangles can be formed with the following side lengths:

• 1. 4 cm, 7 cm, 10 cm
• 2. 24 mm, 20 mm, 30 mm

4 + 7 > 10 24 + 20 > 30 4 + 10 > 7 24 + 30 > 20 7 + 10 > 4 20 + 30 > 24 YES YES

• 3. 7 ft, 9 ft, 16 ft
• 4. 9 in, 13 in, 24 in

7 + 9 = 16 9 + 13 < 24 7 + 16 > 9 9 + 24 > 13 16 + 9 > 7 13 + 24 > 9 NO NO

Click Click Click Click

Try These

1 Determine if sides of length 5 mm, 14 mm and 19 mm can form a triangle. Be prepared to show your work! Yes No

Answer

2 Determine if sides of length 6 in, 9 in and 14 in can form a

• triangle. Be prepared to show your work!

Yes No

Answer

Yes

3 Determine if sides of length 5 yd, 13 yd and 21 yd can form a triangle. Be prepared to show your work! Yes No

Answer

No

4 Determine if sides of length 3 ft, 8 ft and 15 ft can form a

• triangle. Be prepared to show your work!

Yes No

Answer

No

5 Determine if sides of length 5 in, 5 in and 9 in can form a

• triangle. Be prepared to show your work!

Yes No

Answer

Yes

6 A triangle could have which of the following sets of angles? A 40º, 90º, 105º B 35º, 89º, 56º C 75º, 90º, 15º D 30º, 65º, 95º

Answer

B

7 A triangle could have which of the following sets of angles? A 37º, 63º, 80º B 90º, 104º, 76º C 23º, 47º, 50º D 80º, 90º, 10º

Answer

Example

Predict the length of the third side of a triangle with sides of length 12 ft and 16 ft. Side 1 = 12 ft Side 2 = 16 ft The 3rd side must be less than: 12 + 16 > 3rd side 28 ft > 3rd side The 3rd side must be greater than: 12 + 3rd side > 16 3rd side > 4 The 3rd side must be greater than 4 ft and less than 28 ft.

Example

Predict the length of the third side of a triangle with sides of length 9 cm and 15 cm. Side 1 = 9 cm Side 2 = 15 cm The 3rd side must be less than: 9 + 15 > 3rd side 24 cm > 3rd side The 3rd side must be greater than: 9 + 3rd side > 15 3rd side > 6 The 3rd side must be greater than 6 cm and less than 24 cm.

Try These

Predict the length of the third side of a triangle whose known sides are lengths:

• 1. 13 mm, 20 mm
• 2. 7 in, 19 in

13 + 20 > Side 3 7 + 19 > Side 3 33 > Side 3 26 > Side 3 13 + Side 3 > 20 7 + Side 3 > 19 Side 3 > 7 Side 3 > 12 7 < side 3 < 33 12 < side 3 < 26

Click Click

Try These

Predict the length of the third side of a triangle whose known sides are lengths:

• 3. 4 ft, 11 ft
• 4. 23 cm, 34 cm

4 + 11 > Side 3 23 + 34 > Side 3 15 > Side 3 57 > Side 3 4 + Side 3 > 11 23 + Side 3 > 34 Side 3 > 7 Side 3 > 11 7 < side 3 < 15 11 < side 3 < 57

Click Click

8 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 6 m and 12 m.

Answer

9 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 6 m and 12 m.

Answer

18 m

10 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 9 in and 17 in.

Answer

8 in

11 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 9 in and 17 in.

Answer

12 Predict the lower limit of the length of the third side of a triangle whose known sides are lengths 15 ft and 43 ft.

Answer

28 ft

13 Predict the upper limit of the length of the third side of a triangle whose known sides are lengths 15 ft and 43 ft.

Answer

58 ft

Geometric Constructions: The Basics

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• f Contents

In Geometry, we can draw just about every figure with various

• tools. The tools that we will be using are given on this slide & the

next slide: 1) Compass: creating circles & arcs

0° 86

Geometric Tools

Teacher Notes

2) Ruler : measure segments D E DE = 6 cm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 0°

B

C

A

180 10 170 2 1 6 30 150 40 140 50 130 60 120 7 110 80 100 90 90 0° 100 80 110 70 120 60 130 50 140 4 1 5 30 160 20 170 10 180

m∠ABC = 65° 3) Protractor: measure angles

Geometric Tools

Example

Draw a circle that has a radius of 6 cm. Step #1: Draw a segment with the ruler that measures 6 cm. Step #2: Line up your compass so that it's center tip lies on one endpoint & the pencil tip lies on the other endpoint.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 0°

0° 179

Step #3: Keeping the distance between the center & endpoint the same, draw your circle.

0° 179

Teacher Notes

Radius

Construct a circle that has a radius of 3 cm using a ruler & a compass. Teacher Notes

Circulate around the room to make sure that the students are constructing the circles

• correctly. Figures should

resemble the figure below.

Radius

Construct a circle that has a radius of 8 cm using a ruler & a compass. Video: Constructing Circles Teacher Notes

Circulate around the room to make sure that the students are constructing the circles

• correctly. Figures should

resemble the figure below.

A B

Example

Use a ruler & a compass to draw an isosceles triangle with the following conditions:

• 1. at least one of the sides is 7 cm
• 2. at least one of the sides is 3 cm

Step #1: Look at your conditions. Both of them say "at least one" which means that one side, or more sides could meet the conditions. Plus, since the triangle is isosceles, we know that two of the sides must be equal. So pick which number you want to occur for 2 of your sides. I'll select the 7 cm to occur twice. Step #2: Draw one of your 7 cm segments.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 0°

A B Step #3: With your compass, draw a circle or semicircle (whichever you prefer).

0° 211

Any segment that I connect from this arc to the center will have the same radius length of 7 cm.

A B

1 2 3 4 5 6 7 8 9 1 11 1 2 1 3 1 4 15 1 2 3 4 5 9 9 °

Step #5: Connect this point with points A & B to form your triangle.

A B C

Note: AC = AB = 7 cm, since they are both radii of the circle. Step #4: With your ruler, find the segment that can be drawn from B to another point on the semicircle so that the ruler measures 3 cm. Make a point at this location.

Think about this...

Can we make a triangle if we use the 3 cm twice and the 7 cm

• nce?

Discuss this problem in your groups for a few minutes. Answer

Isosceles Triangle

Use a ruler & a compass to draw an isosceles triangle with the following conditions:

• 1. at least one of the sides is 8 cm
• 2. at least one of the sides is 6 cm

Video: Constructing Isosceles Triangles

Answer Circulate around the room to make sure that the students are constructing the triangle

• correctly. Note: 2 answers (could

ask them to find both for extra practice): 8 cm, 8 cm & 6 cm & 6 cm, 6 cm & 8 cm

6 cm

Example

Use a ruler & a protractor to draw an equilateral triangle that has a side length of 5.5 cm. Step #1: Reread your question. We have an equilateral triangle, which has all of the sides measuring 5.5 cm & all of its angles measure to be 60°. Step #2: Draw one of your 5.5 cm segments. C D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 0°

C D

180 10 170 20 1 6 30 150 40 140 50 130 6 120 70 1 1 8 100 9 9 ° 1 8 110 7 1 2 6 130 5 140 40 1 5 30 160 2 1 7 10 180

Step #3: With your protractor, place its center at either point C or point D (doesn't matter), measure 60° & draw a line connecting the 60° measurement with your center (mine is C). Repeat step #3 with your other endpoint as the center.

C D

180 10 170 2 1 6 3 150 40 140 50 130 60 1 2 7 1 1 80 100 90 90 0° 100 8 110 70 1 2 6 130 50 140 40 1 5 30 1 6 2 1 7 10 180

Example

C D Step #4: Draw a point where the 2 lines intersect & erase any additional lines. Note: You can verify that all of the edges are equal by measuring all

• f them.

Example

Equilateral Triangle

Construct an equilateral triangle that has a side length of 4.2 cm using a protractor and a ruler. Answer

Circulate around the room to make sure that the students are constructing the triangles

• correctly. Diagrams should

resemble the figure below.

Equilateral Triangle

Construct an equilateral triangle that has a side length of 6.7 cm using a protractor and a ruler. Video: Constructing Equilateral Triangles Answer

Use a ruler & a protractor to draw ∆ FGH that satisfies the following conditions.

• 1. m∠F = 30°
• 2. m∠G = 70°
• 3. FG = 8 cm

Step #1: Reread your question. We know two angle measurements and the length of 1 side. Let's start with the side, since at least 1 segment is required to use a protractor. Step #2: Draw your 8 cm segment. Label one endpoint with F & the

• ther with G

F G

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 0°

Example

F G

180 1 170 20 160 30 150 40 1 4 50 130 60 120 7 110 80 100 90 90 0° 100 80 110 70 1 2 60 130 5 140 40 1 5 30 160 20 170 10 1 8

Step #2: With your protractor, place its center at point F. Measure the 30° required for m∠F. Draw a line through the 30° angle.

F G

180 10 1 7 20 160 30 150 40 140 50 130 60 120 70 1 1 8 100 90 90 0° 100 8 110 70 1 2 60 130 50 140 4 150 30 160 20 170 10 180

Step #3: With your protractor, place its center at point G. Measure the 70° required for m∠G. Draw a line through the 70° angle.

Example

Step #4: Draw point H where the 2 lines intersect & erase any additional lines. F G H 30° 70°

8 cm

Example

Try This!

Construct ∆JKL with the given conditions using a protractor and a ruler.

• 1. m∠K = 105°
• 2. m∠L = 25°
• 3. KL = 9 cm

Video: Constructing Triangles Answer

Circulate around the room to make sure that the students are constructing the triangles correctly. Their diagrams should resemble the figure below.

Glossary

Teacher Notes

Vocabulary Words are bolded in the presentation. The text box the word is in is then linked to the page at the end

• f the presentation with the

word defined on it.

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• f Contents

Back to Instruction

Compass

An instrument with 2 arms, one sharp &

• ne with a pencil that can be used to

draw circles and arcs.

0° 62 0° 100 0° 48

Back to Instruction

Protractor

An instrument used to measure angles. They are usually half of a circle (180°). To measure an angle, line up the protractor so that the vertex of the angle aligns with the small hole, or dot, near the bottom of the protractor and one of the angle edges lies on the straight line at the bottom.

30°

90°

135°

Back to Instruction

Ruler

An instrument used to measure the lengths of segments.

1 2 3 4 5 6 7 8 1 2 3 3 2 °

A B C D

1 2 3 4 5 6 7 8 1 2 3 27° 1 2 3 4 5 6 7 8 1 2 3 90°

E F

AB = 3.5 cm CD = 5 cm EF = 6.5 cm

Back to Instruction

5 10 3 5 10 3 5 + 3 > 10

Triangle Inequality

The sum of the lengths of any two sides

• f a triangle is greater than the length
• f the third side.

6 10 4 6 10 4 6 + 4 > 10 7 10 5 7 10 5 7 + 5 > 10

Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull­tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull­tab.

Standards for Mathematical Practices