Similar Triangles and Transformations (Grade 10)

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Similar Triangles and Transformations

Let [math]Delta ABC[/math] be the triangle with vertices located at A(0,1), B(3,4), and C(6,-1). The lengths of the sides of the triangle are: [math]AB = 3sqrt(2) \ "units"[/math], [math]BC = sqrt(34) \ "units"[/math], and [math]AC = 2sqrt(10) \ "units"[/math]. The angle measures of the triangle, rounded to 1 decimal place, are: [math]m ang BAC = 63.4°[/math], [math]m ang ABC = 76.0°[/math], and [math]m ang BCA = 40.6°[/math].

Note: For 3 points P, Q, and R, the measure of the angle formed (with Q at the vertex) by these 3 points is given by the following formula:
[math]m ang PQR = cos^{-1} ( (PR^2 - PQ^2 - QR^2)/(-2 \ PQ \ \ QR))[/math].
1. 
[math]Delta ABC[/math] undergoes the following sequence of transformations: a translation of 3 units right and 2 units down, a reflection over the x-axis, and a dilation of factor 3 centered at the origin. Which of the following points represent the vertices of the transformed triangle, [math]Delta A'B'C' ?[/math]
  1. A'(-9,-2), B'(-18,6), and C'(-27,-9)
  2. A'(9,1), B'(18,-3), and C'(27,3)
  3. A'(9,3), B'(18,-6), and C'(27,9)
  4. A'(-6,-12), B'(3,-12), and C'(12,-6)
2. 
What is the length of side [math]bar{A'B'} ?[/math]
  1. [math]9 sqrt(2) \ "units"[/math]
  2. [math]3sqrt(10) \ "units"[/math]
  3. [math]9 \ "units"[/math]
  4. [math]sqrt(97) \ "units"[/math]
3. 
What is the length of side [math]\bar{B'C'} ?[/math]
  1. [math]sqrt(337) \ "units"[/math]
  2. [math]3sqrt(10) \ "units"[/math]
  3. [math]3sqrt(13) \ "units"[/math]
  4. [math]3sqrt(34) \ "units"[/math]
4. 
What is the length of side [math]\bar{A'C'} ?[/math]
  1. [math]sqrt{373} \ "units"[/math]
  2. [math]2sqrt(82) \ "units"[/math]
  3. [math]6sqrt(10) \ "units"[/math]
  4. [math]12sqrt(2) \ "units"[/math]
5. 
What is the measure of [math]ang B'A'C'[/math], rounded to 1 decimal place?
  1. 63.4°
  2. 21.1°
  3. 88.3°
  4. 116.6°
6. 
What is the measure of [math]ang A'B'C'[/math], rounded to 1 decimal place?
  1. 104.0°
  2. 88.6°
  3. 25.3°
  4. 76.0°
7. 
What is the measure of [math]ang A'C'B'[/math], rounded to 1 decimal place?
  1. 82.1°
  2. 40.6°
  3. 13.5°
  4. 121.8°
8. 
Which of the following correctly states the relationship between the side lengths of the pre-image [math]Delta ABC[/math] and the side lengths of the image [math]Delta A'B'C' ?[/math]
  1. [math](A'B')/(AB) = (B'C')/(BC) = (A'C')/(AC)[/math]
  2. [math]A'B' = 2AB, \ B'C' = 2BC, \ A'C' = 2AC[/math]
  3. [math]A'B' + B'C' + A'B' = 1/3(AB+ BC+ AC)[/math]
  4. [math]A'B'^2 = BC^2 + AC^2, \ B'C'^2 = AB^2 +AC^2, \ A'C'^2 = AB^2 + BC^2 [/math]
9. 
Which of the following correctly states the relationship between the angle measures of the pre-image [math]Delta ABC[/math] and the angle measures of the image, [math]Delta A'B'C' ?[/math]
  1. [math]m ang B'A'C' = m ang BAC^2 , \ m ang A'B'C' = m ang ABC ^2, \ m ang A'C'B' = m ang ACB ^2 [/math]
  2. [math]m ang BAC = 2m ang B'A'C', \ m ang ABC = 2m ang A'B'C', \ m ang ACB = 2m ang A'C'B'[/math]
  3. [math]m ang BAC = m ang B'A'C', \ m ang ABC = m ang A'B'C', \ m ang ACB = m ang A'C'B'[/math]
  4. [math]m ang BAC = 1/3 m ang B'A'C', \ m ang ABC = 1/3 m ang A'B'C', \ m ang ACB = 1/3 m ang A'C'B'[/math]
10. 
The previous questions have shown that [math]Delta ABC \ ~ \ Delta A'B'C'[/math]. If additional transformations were added to the sequence of transformations in question 1, (transformations such as another translation, a rotation, another reflection, or another dilation), would this alter the results in the previous two questions?
  1. The results in the previous two questions would only be altered if another dilation were added to the sequence of transformations. Otherwise, there would be no change.
  2. Adding any one of the additional transformations would affect the results in the previous two questions.
  3. None of the additional transformations, if added to the sequence of transformations, would affect the results in the previous two questions.
  4. Adding additional transformations would only affect the results in the previous two questions if they included a transformation not around the origin (such as a rotation about a point that is not the origin, or a dilation not centered at the origin).

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