Question

Question 1 (Essay Worth 10 points)
(02.01, 02.02 HC)

Triangle ABC has vertices A(−3, 1), B(−3, 4), and C(−7, 1).



Part A: If ∆ABC is translated according to the rule (x, y) → (x + 4, y − 3) to form ∆A'B'C', how is the translation described with words? (3 points)
Part B: Where are the vertices of ∆A'B'C' located? Show your work or explain your steps. (4 points)
Part C: Triangle A'B'C' is rotated 90° counterclockwise about the origin to form ∆A"B"C". Is ∆ABC congruent to ∆A"B"C"? Give details to support your answer. (3 points)





Question 2 (Essay Worth 10 points)
(02.03 MC)

Determine if the two figures are congruent and explain your answer using transformations.

Figure ABCD is shown. A is at negative 1, 1. B is at negative 3, 2. C is at negative 4, 4. D is at 0, 5. Figure EFGH is shown. E is at 1, 1. F is at 3, 0. G is at 4, negative 2. H is at 0, negative 3.





Question 3 (Essay Worth 10 points)
(02.04 LC)

In triangles GHI and RST, ∠G ≅ ∠R, ∠H ≅ ∠S, and segment GI ≅ segment RT. Is this information sufficient to prove triangles GHI and RST congruent through SSS? Explain your answer.





Question 4 (Essay Worth 10 points)
(02.06 MC)

Quadrilateral RSTU, diagonals SU and RT intersect at point V.

RSTU is a parallelogram. If m∠TSV = 31° and m∠SVT = 126°, explain how you can find the measure of ∠URV. Show all steps of your work, and refer to any properties of triangles, parallelograms, or triangle congruency theorems as necessary to justify your response.

Answer 1

### Question 1

**Part A:**
The translation rule \((x, y) \to (x + 4, y - 3)\) describes moving each point of triangle ABC four units to the right (positive x-direction) and three units down (negative y-direction). In words, the translation shifts the entire triangle four units to the right and three units downward.

**Part B:**
To find the new vertices of triangle A'B'C' after the translation, we apply the translation rule to each vertex of triangle ABC:

- For vertex A(-3, 1):
- A': \((-3 + 4, 1 - 3) = (1, -2)\)

- For vertex B(-3, 4):
- B': \((-3 + 4, 4 - 3) = (1, 1)\)

- For vertex C(-7, 1):
- C': \((-7 + 4, 1 - 3) = (-3, -2)\)

Thus, the vertices of triangle A'B'C' are A'(1, -2), B'(1, 1), and C'(-3, -2).

**Part C:**
Triangle A'B'C' is rotated 90° counterclockwise about the origin. To determine whether triangles ABC and A"B"C" are congruent, we first analyze the transformation of each vertex through the 90° rotation.

- Rotating A'(1, -2):
- New position after rotation: A" = (2, 1)

- Rotating B'(1, 1):
- New position after rotation: B" = (-1, 1)

- Rotating C'(-3, -2):
- New position after rotation: C" = (2, -3)

The transformed triangle A"B"C" retains the same structure and size as A'B'C' and therefore, all corresponding sides and angles remain unchanged. Thus, triangles ABC and A"B"C" are congruent due to the properties of rigid transformations (specifically translations and rotations both preserving distances and angles).

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### Question 2

To determine if figures ABCD and EFGH are congruent, we first analyze their vertices:

- Figure ABCD:
- A(-1, 1)
- B(-3, 2)
- C(-4, 4)
- D(0, 5)

- Figure EFGH:
- E(1, 1)
- F(3, 0)
- G(4, -2)
- H(0, -3)

To check for congruence using transformations, we consider translation and rotation. We compare the lengths of the sides of both quadrilaterals:

1. **Calculate Side Lengths:**
- **For ABCD:**
- AB = √((-3 - (-1))^2 + (2 - 1)^2) = √(4 + 1) = √5
- BC = √((-4 - (-3))^2 + (4 - 2)^2) = √(1 + 4) = √5
- CD = √((0 - (-4))^2 + (5 - 4)^2) = √(16 + 1) = √17
- DA = √((-1 - 0)^2 + (1 - 5)^2) = √(1 + 16) = √17

- **For EFGH:**
- EF = √((3 - 1)^2 + (0 - 1)^2) = √(4 + 1) = √5
- FG = √((4 - 3)^2 + (-2 - 0)^2) = √(1 + 4) = √5
- GH = √((0 - 4)^2 + (-3 - (-2))^2) = √(16 + 1) = √17
- HE = √((1 - 0)^2 + (1 - (-3))^2) = √(1 + 16) = √17

Since the lengths of sides correspond (AB = EF, BC = FG, CD = GH, DA = HE), and if the angles between the corresponding sides are also the same, we would conclude that figures ABCD and EFGH are congruent.

Therefore, based on this analysis, **the figures are congruent** as we've verified side lengths, and the angles can be shown to match through translations and possible rotations.

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### Question 3

In triangles GHI and RST, we are given the following information:

- ∠G ≅ ∠R
- ∠H ≅ ∠S
- Segment GI ≅ Segment RT

This means we have two angles and a non-included side corresponding between the triangles. However, to use the Side-Side-Side (SSS) theorem for congruence, we require three pairs of corresponding sides to be congruent. Here, we do not have that since we only have one pair of sides that are congruent, along with angles.

Thus, we cannot conclude triangle congruence through SSS, but rather through the Angle-Angle-Side (AAS) or Angle-Side-Angle (ASA) congruence criteria if we had the third side of one triangle matching the corresponding third side of the other, which we do not know.

**In conclusion**, the information provided is **not sufficient** to prove triangles GHI and RST congruent through SSS; at least three pairs of sides must be known as congruent in order to apply the SSS theorem.

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### Question 4

Given that quadrilateral RSTU is a parallelogram and that diagonals SU and RT intersect at point V, we know several properties of parallelograms. Specifically, in a parallelogram, opposite angles are equal, and adjacent angles are supplementary (sum to 180°).

We are given:
- m∠TSV = 31°
- m∠SVT = 126°

To find the measure of ∠URV, we can first calculate m∠TVS since V is the intersection point of diagonals in a parallelogram.

Knowing that:
m∠TSV + m∠SVT + m∠TVS = 180° (angles on a straight line)

Substituting the known values:
31° + 126° + m∠TVS = 180°
m∠TVS = 180° - 157°
m∠TVS = 23°

Since opposite angles in a parallelogram are congruent:
- m∠URV = m∠TVS = 23°

Thus, the measure of ∠URV is 23°.

**To summarize the steps:**
1. Calculate missing angles using the property of straight lines (sum to 180°).
2. Reuse the parallelogram property that opposite angles are equal.

So the conclusion is that m∠URV = 23°.