# 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].

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]

- A'(-9,-2), B'(-18,6), and C'(-27,-9)
- A'(9,1), B'(18,-3), and C'(27,3)
- A'(9,3), B'(18,-6), and C'(27,9)
- A'(-6,-12), B'(3,-12), and C'(12,-6)

2.

What is the length of side [math]bar{A'B'} ?[/math]

- [math]9 sqrt(2) \ "units"[/math]
- [math]3sqrt(10) \ "units"[/math]
- [math]9 \ "units"[/math]
- [math]sqrt(97) \ "units"[/math]

3.

What is the length of side [math]\bar{B'C'} ?[/math]

- [math]sqrt(337) \ "units"[/math]
- [math]3sqrt(10) \ "units"[/math]
- [math]3sqrt(13) \ "units"[/math]
- [math]3sqrt(34) \ "units"[/math]

4.

What is the length of side [math]\bar{A'C'} ?[/math]

- [math]sqrt{373} \ "units"[/math]
- [math]2sqrt(82) \ "units"[/math]
- [math]6sqrt(10) \ "units"[/math]
- [math]12sqrt(2) \ "units"[/math]

5.

What is the measure of [math]ang B'A'C'[/math], rounded to 1 decimal place?

- 63.4°
- 21.1°
- 88.3°
- 116.6°

6.

What is the measure of [math]ang A'B'C'[/math], rounded to 1 decimal place?

- 104.0°
- 88.6°
- 25.3°
- 76.0°

7.

What is the measure of [math]ang A'C'B'[/math], rounded to 1 decimal place?

- 82.1°
- 40.6°
- 13.5°
- 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]

- [math](A'B')/(AB) = (B'C')/(BC) = (A'C')/(AC)[/math]
- [math]A'B' = 2AB, \ B'C' = 2BC, \ A'C' = 2AC[/math]
- [math]A'B' + B'C' + A'B' = 1/3(AB+ BC+ AC)[/math]
- [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]

- [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]
- [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]
- [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]
- [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?

- 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.
- Adding any one of the additional transformations would affect the results in the previous two questions.
- None of the additional transformations, if added to the sequence of transformations, would affect the results in the previous two questions.
- 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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