Triangle Congruence from Rigid Motions
Help Questions
Geometry › Triangle Congruence from Rigid Motions
Triangle $$PQR$$ with vertices $$P(3, 4)$$, $$Q(7, 2)$$, $$R(5, 8)$$ is congruent to triangle $$STU$$. If the congruence can be established through rigid motions, but triangle $$STU$$ has vertices $$S(-4, 3)$$, $$T(-2, 7)$$, $$U(-8, 5)$$, what must be verified to confirm the triangles are indeed congruent?
Check that corresponding angles are equal and that no dilations were used in the transformation
Confirm that the triangles have the same perimeter and the same area measurements
Calculate the side lengths of both triangles and verify they form the same set of distances
Verify that one triangle can be mapped onto the other using only translations and rotations
Explanation
When determining congruence between triangles using coordinate geometry, you need to verify that corresponding sides have identical lengths. Congruent triangles have exactly the same shape and size, which means all corresponding sides must be equal.
For triangle $$PQR$$, calculate the side lengths using the distance formula $$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$:
- $$PQ = \sqrt{(7-3)^2 + (2-4)^2} = \sqrt{16+4} = \sqrt{20}$$
- $$QR = \sqrt{(5-7)^2 + (8-2)^2} = \sqrt{4+36} = \sqrt{40}$$
- $$PR = \sqrt{(5-3)^2 + (8-4)^2} = \sqrt{4+16} = \sqrt{20}$$
For triangle $$STU$$:
- $$ST = \sqrt{(-2-(-4))^2 + (7-3)^2} = \sqrt{4+16} = \sqrt{20}$$
- $$TU = \sqrt{(-8-(-2))^2 + (5-7)^2} = \sqrt{36+4} = \sqrt{40}$$
- $$SU = \sqrt{(-8-(-4))^2 + (5-3)^2} = \sqrt{16+4} = \sqrt{20}$$
Both triangles have the same set of side lengths: $${\sqrt{20}, \sqrt{20}, \sqrt{40}}$$, confirming congruence by SSS.
Option A is incorrect because checking angles isn't necessary when you can prove SSS congruence, and mentioning dilations is irrelevant. Option B is wrong because equal perimeter and area don't guarantee congruence—different triangles can share these measurements. Option C is incomplete because it doesn't specify what needs verification about the mapping.
Strategy tip: For coordinate geometry congruence problems, always calculate and compare corresponding side lengths first. The SSS congruence test is the most direct method when working with coordinates.
Triangle $$RST$$ is congruent to triangle $$UVW$$ with the correspondence $$R \leftrightarrow U$$, $$S \leftrightarrow V$$, $$T \leftrightarrow W$$. If a $$270°$$ counterclockwise rotation about point $$P$$ maps triangle $$RST$$ onto triangle $$UVW$$, which statement about the angle measures is necessarily true?
Each angle in triangle $$UVW$$ is $$270°$$ greater than the corresponding angle in triangle $$RST$$
The sum of corresponding angles from both triangles equals $$360°$$ for each pair
The angles in triangle $$UVW$$ are the supplements of the corresponding angles in triangle $$RST$$
$$\angle R = \angle U$$, $$\angle S = \angle V$$, and $$\angle T = \angle W$$ regardless of the rotation
Explanation
Rigid motions, including rotations, preserve angle measures. Therefore, corresponding angles are congruent regardless of the specific rotation applied. The 270° rotation affects the position and orientation of the triangle, not the individual angle measures within it. Choices A, B, and D incorrectly suggest that the rotation angle affects the triangle's internal angle measures.
Triangle $$JKL$$ undergoes a rigid motion to produce triangle $$MNO$$. If $$JK = 15$$, $$KL = 9$$, $$JL = 12$$, and $$MN = 9$$, $$NO = 12$$, $$OM = 15$$, which additional information is needed to conclude that the triangles are congruent?
No additional information is needed since corresponding sides are already congruent
The specific type of rigid motion (translation, rotation, or reflection) must be identified
The correspondence between vertices must be established to verify angle congruence
At least one pair of corresponding angles must be measured and verified equal
Explanation
While all corresponding sides are equal in length (SSS), we need to establish which vertices correspond to conclude the triangles are congruent. The given side lengths could correspond in multiple ways (J↔M, K↔N, L↔O or other arrangements). Choice B is unnecessary since SSS guarantees angle congruence. Choice C is irrelevant for proving congruence. Choice D ignores the correspondence issue.
Triangle $$ABC$$ is congruent to triangle $$DEF$$ by a sequence of rigid motions. Given that $$AB = 13$$ cm, $$BC = 8$$ cm, $$AC = 15$$ cm, and in triangle $$DEF$$: $$DE = 8$$ cm, $$EF = 15$$ cm, $$DF = 13$$ cm, which correspondence between vertices ensures the congruence?
Any correspondence works since all sides are different lengths in each triangle
$$A \leftrightarrow F$$, $$B \leftrightarrow D$$, $$C \leftrightarrow E$$ based on matching side lengths
$$A \leftrightarrow D$$, $$B \leftrightarrow E$$, $$C \leftrightarrow F$$ with all sides matching in order
$$A \leftrightarrow E$$, $$B \leftrightarrow F$$, $$C \leftrightarrow D$$ to align equal sides properly
Explanation
We must match corresponding sides: AB = 13 matches DF = 13, so A↔F and B↔D. BC = 8 matches DE = 8, so B↔D and C↔E. AC = 15 matches EF = 15, so A↔F and C↔E. This gives correspondence A↔F, B↔D, C↔E. Choice A incorrectly matches sides in order. Choice C creates wrong pairings. Choice D ignores the need for proper correspondence.
In triangle $$DEF$$, $$DE = 7$$ cm, $$EF = 10$$ cm, and $$\angle E = 65°$$. Triangle $$GHI$$ has $$GH = 10$$ cm, $$HI = 7$$ cm, and $$\angle H = 65°$$. A student claims these triangles are congruent by SAS and can be mapped onto each other by rigid motions. Which analysis of this claim is correct?
The claim is incorrect because the sides are not in the same order relative to the given angle
The claim is incorrect because rigid motions require SSS congruence, not SAS congruence
The claim is correct because both triangles have two equal sides and an equal included angle
The claim is correct because SAS congruence guarantees the existence of a rigid motion mapping
Explanation
The triangles satisfy SAS congruence: DE = HI = 7 cm, EF = GH = 10 cm, and ∠E = ∠H = 65°. When two triangles are congruent by any valid method (SAS, SSS, ASA, AAS), there exists a sequence of rigid motions that maps one onto the other. Choice A misses the correspondence verification. Choice B incorrectly suggests the sides aren't properly arranged. Choice D incorrectly states that only SSS works with rigid motions.