SSS, SAS, and ASA and Rigid Transformations (Grade 10)

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SSS, SAS, and ASA and Rigid Transformations

1. 
You are given two triangles, ABC and DEF. If you know that [math]bar{AB}~=bar{DE}, bar{BC}~=bar{EF}[/math] and [math]ang B ~= ang E[/math]. Which of the following could be true? Select all that apply.
  1. There is not enough information to make any conclusions.
  2. A dilation and translation map triangle ABC onto triangle DEF.
  3. A translation and rotation map triangle ABC onto triangle DEF.
  4. A rotation and reflection map triangle ABC onto triangle DEF.
2. 
You are given two triangles, ABC and GHF, plotted on a set of axes. You know that you can map [math]bar{AB}[/math] onto [math]bar{GH}[/math] by a reflection over the x-axis and a rotation of 160° clockwise about the origin. You also know that [math]ang A ~= ang G[/math] and [math]ang B ~= ang H[/math]. Which of the following statements are true? There may be more than one correct answer.
  1. [math]Delta ABC ~= Delta GHF[/math]
  2. [math]bar{BC}[/math] can be mapped onto [math]bar{HF}[/math] by a reflection over the x-axis and a rotation of 160° clockwise about the origin.
  3. [math]bar{AC}[/math] can be mapped onto [math]bar{GF}[/math] by a reflection over the x-axis and a rotation of 160° clockwise about the origin.
  4. [math]m ang C = m ang F + 160°[/math]
3. 
You are given two triangles, ABC and LMN. The vertices of triangle ABC are [math](-3,2), (-1,2), (-2,4)[/math]. The vertices of triangle LMN are [math](4,-2), (2,-2), (3,-4).[/math]
A. 
Which of the following sequences of transformations would map [math]bar{AB}[/math] onto [math]bar{LM}[/math], such that point A corresponds with point L and point B with point M?
  1. Reflection over the y-axis, translation of one unit right, reflection over the x-axis.
  2. Reflection over the x-axis, translation 7 units right.
  3. Rotation of 90° about the origin, translation 3 units down.
  4. Reflection over the line y = x, translation 7 units right, translation 4 units down.
B. 
Would the sequence of transformations that mapped [math]bar{AB}[/math] onto [math]bar{LM}[/math] be the same for mapping [math]bar{AC}[/math] onto [math]bar{LN}[/math] and [math]bar{BC}[/math] onto [math]bar{MN} ?[/math]
  1. No.
  2. Only for mapping [math]bar{AC}[/math] onto [math]bar{LN}[/math].
  3. Only for mapping [math]bar{BC}[/math] onto [math]bar{NM} [/math].
  4. Yes.
C. 
Because the three sides of ABC can be mapped by the same sequence of rigid transformations to the three sides of triangle LMN, what does this imply and why?
  1. [math]Delta ABC ~= Delta LMN[/math], since no rotations were used.
  2. [math]Delta ABC ~= Delta LMN[/math], since rigid transformations preserve size and shape.
  3. The triangles are not necessarily congruent, since we know nothing about the angles.
  4. The triangles are not necessarily congruent, since each side was considered separately.
D. 
From parts A and B, which of the following triangle congruence theorems could be used to show [math]Delta ABC ~= Delta LMN ?[/math]
  1. SAS
  2. ASA
  3. SSS
  4. AAS
4. 
Points P, Q, and R are located at (2, 1), (1, 4), (4, 5).
A. 
What is the measure of [math]ang PQR ?[/math]
  1. 30°
  2. 90°
  3. 180°
B. 
What are the coordinates of the transformed points, P', Q', R', if they are rotated 90° clockwise about the origin and then translated 4 units to the left?
  1. (-3, -2), (0, -1), (1, -4)
  2. (-5, 2), (-8, 1), (-9, 4)
  3. (1, 2), (4, 3), (5, 0)
  4. (-6, -1), (-5, -4), (-8, -5)
C. 
What is the measure of [math]ang P'Q'R' ?[/math]
  1. 90°
  2. 180°
  3. 60°
D. 
Aside from calculating the length of each line segment, what reasoning can be used to conclude that [math]bar{PQ}~= bar{P'Q'}[/math] and [math]bar{QR} ~= bar{Q'R'} ?[/math]
  1. They have similar names.
  2. The points which define them were transformed by the same rigid transformations.
  3. Reflections were not involved in the transformations.
  4. They are specified by Cartesian coordinates.
E. 
We know that [math]Delta PQR ~= Delta P'Q'R'[/math], due to the points defining the triangles being transformed by the same sequence of rigid transformations. However, if we did not know the coordinates of these points, and only knew the information of parts A, C, and D, could we still conclude that [math]Delta PQR ~= Delta P'Q'R'[/math], and why?
  1. No, there would not be enough information to make this conclusion.
  2. No, because then the two triangles could have been transformed by other transformations.
  3. Yes, because of the SSS theorem.
  4. Yes, because of the SAS theorem.

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