Congruent Triangles and Rigid Motion Name ______
Directions: The following problems deal with congruency and rigid motion. The term “rigid motion” is also known as “isometry” or “congruence transformations.”
1. In the diagram at the right, a transformation has occurred on DABC.
a) Describe a transformation that created image DA¢B¢C¢ from DABC.
Translation: (x, y) ® (x – 6, y – 7)
b) Is DABC congruent to DA¢B¢C¢? ______Explain.
A translation is a rigid motion which preserves
length, thus creating a figure with the same size
and shape as the pre-image.
2. The vertices of DMAP are M(-8, 4), A(-6, 8) and P(-2, 7).
The vertices of DM¢A¢P¢ are M¢(8, -4), A¢(6, -8) and P¢(2, -7).
a) Plot DM¢A¢P¢.
b) Verify that the triangles are congruent.
MA = (-8-(-6))2+(4-8)2 = 4+16= 20
M¢A¢ = (8-6)2+(-4-(-8)2 = 4+16= 20
AP = (-6--2)2+ (8-7)2 = 16+1= 17
A¢P¢ = (6-2)2+ (-8--7)2= 16+1= 17
MP = (-8--2)2+ (4-7)2 = 36+9= 45
M¢P¢ = (8-2)2+ (-4--7)2= 36+9= 45
c) Describe a rigid motion that can be used to M¢A¢P¢.
A counterclockwise rotation of 180o.
3. Given DPQR with P(-4, 2), Q(2, 6) and R(0, 0) is congruent
to DSTR with S(2, -4), T(6, 2) and R(0, 0).
a) Plot DSTR.
b) Describe a rigid motion which can be used to verify
the triangles are congruent.
Reflection in line y = x.
4. Given DRST with R(1, 1), S(4, 5) and T(7, 5).
a) Plot the reflection of DRST in the y-axis and label it DR¢S¢T¢.
b) Is DRST congruent to DR¢S¢T¢? __yes__ Explain.
A reflection is a rigid motion which preserves length,
thus creating a figure with the same size and shape as
the pre-image.
c) Plot the image of DR¢S¢T¢under the translation
(x, y) ® (x + 4, y – 8). Label the image of DR¢¢S¢¢T¢¢.
d) Is DR¢S¢T¢ congruent to DR¢¢S¢¢T¢¢? __yes___ Explain.
A translation is a rigid motion which preserves length,
thus creating a figure with the same size and shape as
the pre-image.
e) Is DRST congruent to DR¢¢S¢¢T¢¢? __yes__ Explain. Both triangles are congruent to DR¢S¢T¢, therefore by the transitive property of congruence, they are congruent to each other. Or rigid motion congruence is transitive.
5. Given DDFE with D(1, -1), F(9, 6) and E(5,7) and DBAT
with B(1, 1), A(-6, 9) and T(-7, 5).
a) Describe a transformation that will yield DBAT as
the image of DDFE.
Counterclockwise rotation of 90o
around the origin.
b) Is DBAT congruent to DDF? __yes__ Explain.
A rotation is a rigid motion which preserves length,
thus creating a figure with the same size and shape as
the pre-image.
6. Given DCAP with C(-4, -2), A(2, 4) and P(4, 0) and DSUN
with S(-8, -4), U(4, 8) and N(8, 0).
a) Plot DCAP and DSUN.
b) Describe a transformation that will yield DSUN as
the image of DCAP. A dilation centered at (0,0)
with a scale factor 2.
c) Is DCAP congruent to DSUN? __yes__ Explain.
While these triangles are the same shape, they are not the same size.
A dilation is NOT a rigid motion and does not preserve the congruency of figures.
These triangles are similar, but not congruent.