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Private: Learning Math: Geometry
Triangles and Quadrilaterals Part B: Linkage-Strip Constructions (40 minutes)
Session 2, Part B
In this part:
- Constructing Triangles
- Constructing Quadrilaterals
- Properties of Triangles.
A triangle has three sides, but not just any set of three lengths will make a triangle. Use linkage-strips to answer Problems B1-B5. Note 3
Constructing Triangles
Problem B1
Fill in the table below. Try to build triangles with the given lengths. Write “yes” or “no” in the fourth column of the table to indicate whether you can or cannot make a triangle from those three lengths. Experiment with different sets of lengths. When you build a triangle, see if you can deform it (change its shape) into a different triangle while keeping the side lengths the same. Scroll down the page to Solutions to see the filled-in table. Note 4
|
Side B |
Side C |
Is it a triangle? |
Can it be deformed? |
|
| 4 units | 4 units | 4 units | ||
| 4 units | 3 units | 2 units | ||
| 3 units | 2 units | 1 units | ||
Problem B2
Suppose you were asked to make a triangle with sides 4, 4, and 10 units long. Do you think you could do it? Explain your answer. Keep in mind the goal is not to try to build the triangle, but to predict the outcome.
Problem B3
Come up with a rule that describes when three lengths will make a triangle and when they will not. Write down the rule in your own words.
Problem B4
Suppose you were asked to make a triangle with sides 13.2, 22.333, and 16.5 units long. Do you think you could do it? Explain your answer.
 |
Video SegmentIn this video segment, Vicky and Lolita write a rule that describes when three lengths will make a triangle. Watch this segment after you have completed Problems B1-B4, and compare your rule with that of the onscreen participants. What was the first rule that Vicky wrote? How did she and Lolita revise this rule? How does this rule compare with the one that you wrote? You can find this segment on the session video approximately 9 minutes and 25 seconds after the Annenberg Media logo. |
 |
Video SegmentIn this video segment, Kent describes a different rule for when three lengths will make a triangle. Watch this segment after you have completed Problems B1-B4, and compare your rule with Kent’s. What was Kent’s rule? How is it different from Vicky and Lolita’s rule? How does this rule compare with the one that you wrote? You can find this segment on the session video approximately 12 minutes and 50 seconds after the Annenberg Media logo. |
To answer this question, you will need to know what it means for two triangles to be “different.” One definition says that triangles that are “different” cannot have the exact same size and shape. Rotating or reflecting a triangle with the same size and shape does not produce a “different” triangle.
Problem B5
Can a set of three lengths make two different triangles?
Constructing Quadrilaterals
For this activity, use linkage strips or make your own. See Note 1.
Problem B6
Fill in the table below. Use this linkage-strip Interactive Activity (or the hands-on version) to try to build quadrilaterals with the given lengths. Write “yes” or “no” in the fifth column of the table to indicate whether or not you can make a quadrilateral from those four lengths. Experiment with different sets of lengths. When you build a quadrilateral, see if you can deform it into a different quadrilateral with the same side lengths. When you click “Show Answers,” the filled-in table will appear below the problem. Scroll down the page to see it.
|
Side A |
Side B |
Side C |
Side D |
Is it a quadri- |
Can it be deformed? |
| 4 units | 4 units | 4 units | 4 units | ||
| 4 units | 3 units | 2 units | 2 units | ||
| 3 units | 2 units | 1 unit | 1 unit | ||
| 4 units | 1 unit | 2 units | 1 unit | ||
Problem B7
For some of the lengths above, can you connect them in a different order to make a different quadrilateral? If so, which ones? How is this different from building triangles?
Problem B8
Come up with a rule that describes when four lengths will make a quadrilateral and when they will not. Write down the rule in your own words. (You may want to try some more cases to test your rule.)
Problem B9
Can a set of four lengths make two different quadrilaterals?
Linkage-strip problems adapted from IMPACT Mathematics Course, 1, developed by Education Development Center, Inc. pp. 55-56, © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math
Properties of Triangles
The triangle inequality is a famous result in mathematics. It says that for three lengths to make a triangle, the sum of any two sides must be greater than the third side. Often you will see a picture like this, where a, b, and c represent the three lengths of the sides.
The triangle inequality is the mathematical statement of the old adage, “The shortest distance between two points is a straight line.” If you don’t travel along the straight line, you travel two sides of a triangle, and that trip takes longer.
You have also probably found that triangles are rigid. That is, if a set of lengths makes a triangle, only one triangle is possible. You can’t push on the vertices to make a different triangle with the same three sides. Triangles are the only rigid polygon, which makes them quite useful for construction.
This property is often abbreviated as SSS (side-side-side) congruence. If the three sides of one triangle have the same lengths as the three sides of another triangle, then the two triangles are congruent. That is, they have exactly the same size and shape. All of the angle measurements will match, as will other measurements, such as their areas, the lengths of the corresponding altitudes, and so on. If you cut the two triangles out from a piece of paper, you could fit one exactly on top of the other.
Notes
Note 3
Use linkage strips (or make your own strips) to answer Problems B1-B5.
Note 4
If you are working in groups to construct triangles and then, in Problem B6, quadrilaterals, discuss the rules you came up with for instances in which three lengths make a triangle and in which four lengths make a quadrilateral.
Sometimes people may be confused about the question of whether or not you can “deform” one triangle into anther one. It helps to build another figure — use five sides so as not to give away the answers for the quadrilaterals in Problem B6 — and see how easy it is to push on the sides and angles to form many different pentagons with the same five sides. Then, although the materials may allow a little bit of “wiggle,” it becomes clear whether or not you can deform a triangle into a completely different one.
Solutions
Problem B1
Here is the table filled in for the triangles with given lengths and for other sample triangles.
|
Side A |
Side B |
Side C |
Is it a triangle? |
Can it be deformed? |
| 4 | 4 | 4 |
Yes |
No |
| 4 | 3 | 2 |
Yes |
No |
| 3 | 2 | 1 |
No |
N/A |
|
4 |
3 |
2 |
Yes |
No |
|
1 |
2 |
4 |
No |
N/A |
|
2 |
4 |
4 |
Yes |
No |
|
3 |
1 |
1 |
No |
N/A |
|
2 |
3 |
3 |
Yes |
No |
|
2 |
4 |
2 |
No |
N/A |
Other answers will vary individually, but no triangle will be deformable.
Problem B2
No, this is not possible. If we attach the two sides of lengths of 4 units to the endpoints of the side of length 10, the first two sides will not meet at a point to create a triangle. Together they are too short.
Problem B3
Three lengths can form a triangle only if the sum of the lengths of any two sides is greater than the length of the third side.
Problem B4
Yes. Because the sum of lengths of any two sides is greater than the length of the remaining side, the two sides will be able to meet at a point and create a triangle when attached to the endpoints of the third side.
Problem B5
No, three fixed lengths determine one and only one triangle. This is demonstrated by the fact that none of the triangles found in Problem B1 can be “deformed” into a different shape.
Problem B6
Here is the table filled in for the quadrilaterals with given lengths and other sample quadrilaterals.
|
Side A |
Side B |
Side C |
Side D |
Is it a quadri- |
Can it be deformed? |
| 4 | 4 | 4 | 4 |
Yes |
Yes |
| 4 | 3 | 2 | 2 |
Yes |
Yes |
| 3 | 2 | 1 | 1 |
Yes |
Yes |
| 4 | 1 | 2 | 1 |
No |
N/A |
|
1 |
1 |
1 |
4 |
No |
N/A |
|
2 |
2 |
2 |
2 |
Yes |
Yes |
|
1 |
4 |
3 |
1 |
Yes |
Yes |
|
1 |
3 |
3 |
4 |
Yes |
Yes |
|
2 |
3 |
4 |
1 |
Yes |
Yes |
|
4 |
1 |
1 |
2 |
No |
N/A |
Other answers will vary individually, but all quadrilaterals will be deformable.
Problem B7
As long as no more than two sides of a quadrilateral are equal in length, we can reorder the way the sides are connected and obtain a different quadrilateral. This is not the case with triangles: If we reorder the sides, we get the same triangle.
Problem B8
Four lengths can form a quadrilateral as long as the sum of the lengths of any three sides is greater than the length of the fourth side.
Problem B9
Yes. For example:
Also, if the sides are not the same length, ordering them differently will produce different quadrilaterals.
