TRIANGLES ON A LATTICE
All of the problems below will refer to the nine points (A, B, C, D, E, F, G, H, I) of the square lattice in the figures below. Each triangle that is to be constructed will use three of these nine points as vertices.
1. Using the lattices at the bottom of the next page, draw and color the following triangles, then carefully cut out each of them: ABD, CIG, ACD, GEI.
2. Use the LEFT lattice above.
a. Draw, then color, the triangle with vertices A,B,D.
b. Draw, then fill in with a different color, all triangles that can be found by translating triangle ABD.
c. Draw, then fill in with a third color, the triangles that can be obtained from those in parts a and b by reflection across the longest side of each of the triangles.
3. Use the RIGHT lattice above.
a. Draw, then fill in, triangle EFI.
b. Draw, then fill in with a different color, the triangle that results when EFI is rotated 90o degrees CLOCKWISE using point E as the center of rotation.
c. Draw, then fill in with a third color, the triangle that results when EFI is rotated 90o COUNTER-CLOCKWISE using point E as the center of rotation.
d. Draw, then fill in with a fourth color, the triangle that results when EFI is rotated 180o CLOCKWISE using point E as the center of rotation.
4. Determine the number of triangles in the lattice that have the same size and shape as ABD.
5. How many triangles have the same size and shape as triangle CIG? Describe transformations (translation, reflection, rotation) that begin with CIG and end with each of these triangles.
6. Repeat #5 beginning with triangle GEI.
7. Indicate whether each of the pair of triangles are congruent. If they are congruent describe a sequence of transformations that move the first into the second. If they are not congruent, explain clearly.
a. Triangle ABG and triangle ACF.
b. Triangle ABG and triangle CBG.
c. Triangle ABG and triangle BCI.
8. Indicate eight distinct triangles, no two of them are congruent. Four of them have been investigated in prior exercises. Indicate for each how many triangles exist in the lattice.
9. How many combinations of three lattice points do NOT produce a triangle?
TEACHERS' GUIDE AND NOTES
1. The purpose of cutting out some triangles is to use them as manipulatives in subsequent tasks.
2b. There are three others generated by translations: BCE, DEG, EFH
2c. There are an additional four triangles: BDE, CEF, EGH, FHI.
3b. Result is EGH 3c. Result is BEF 3d. Result is EHG.
4. Answer is 16 -- the 12 generated in exercises 1,2,3 plus the "empty" locations in #3.
5. There are 4 including CIG. Each can be obtained by rotating around E.
6. There are 8 including GEI.
7a. The triangles are congruent. One way: rotate ABG 90o clockwise about A; then translate
7b. The triangles are not congruent. The longest sides are not of equal length.
7c. The triangles are congruent. One way: reflect about the line segment BH.
Problem 7a Problem 7b Problem 7c
8. There are some that have a right angle: ABD (16), ABG (16), ACG (4), BDF (8).
There are some that are acute and isosceles: GBI (4), AHF(4)
And others: ABF(16), ABI(8).
9. There are 8 combinations that produce straight lines, not triangles. The total number of triangles is 76; this is consistent with the number of ways to choose three vertices out of nine (9*8*7/1*2*3=84), then deducting the 8 straight lines.
TRIANGLES ON A LATTICE
1. Is triangle AGE the same as triangle GEA?
2. What is the result of triangle DEG being rotated by 90 degrees counter-clockwise around the point E?
3. What is the result of triangle DEG being rotated by 90 degrees counter-clockwise around the point D?
4. What is the result of triangle ADF being reflected across the line segment DF?
5. What is the result of triangle ACE being translated one unit down, then rotated 90 degrees clockwise around the point E?
6. What is the result of triangle BCG being rotated 360 degrees clockwise around the point B.
7. Determine (and explain) the number of triangles (in the 3x3 lattice) that are congruent to ABG.
8. Determine the number of triangles that satisfy the following: an isosceles right triangle with legs three units long in a 4x4 square lattice.
SUMMATIVE EVALUATION (with answers)
The teacher will ask the students to answer each of the following by themselves (on paper).
1. Is triangle AGE the same as triangle GEA? ANSWER: Yes. The order of vertices does not matter.
2. What is the result of triangle DEG being rotated by 90 degrees counter-clockwise around the point E? ANSWER: Triangle EHI
3. What is the result of triangle DEG being rotated by 90 degrees counter-clockwise around the point D? ANSWER: Triangle ADE.
4. What is the result of triangle ADF being reflected across the line segment DF?
ANSWER: Triangle GDF.
5. What is the result of triangle ACE being translated one unit down, then rotated 90 degrees clockwise around the point E? ANSWER: Triangle BDH.
6. What is the result of triangle BCG being rotated 360 degrees clockwise around the point B.
ANSWER: Triangle BCG since a 360 degree rotation results in no change.
7. Determine (and explain) the number of triangles (in the 3x3 lattice) that are congruent to ABG.
ANSWER: There are 16. There are several ways to explain the strategy used to identify them all.
8. Determine the number of triangles that satisfy the following: an isosceles right triangle with legs three units long in a 4x4 square lattice. ANSWER: There are 4.