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Math  /  Geometry

Question3. QUILTING In the quilt design, assume that angles and segments that appear to be congruent are congruent. Indicate which triangles are congruent.
4. The vertices of △SUV\triangle S U V and △S′U′V′\triangle S^{\prime} U^{\prime} V^{\prime} are S(0,4),U(0,0)S(0,4), U(0,0), V(2,2),S′(0,−4),U′(0,0)V(2,2), S^{\prime}(0,-4), U^{\prime}(0,0), and V′(−2,−2)V^{\prime}(-2,-2). Verify that the triangles are congruent and then name the congruence transformation.
5. The vertices of △QRT\triangle Q R T and △Q′R′T′\triangle Q^{\prime} R^{\prime} T^{\prime} are Q(−4,3),Q′(4,3),R(−4,−2)Q(-4,3), Q^{\prime}(4,3), R(-4,-2), R′(4,−2),T(−1,−2)R^{\prime}(4,-2), T(-1,-2), and T′(1,−2)T^{\prime}(1,-2). Verify that △QRT≅△Q′R′T′\triangle Q R T \cong \triangle Q^{\prime} R^{\prime} T^{\prime}. Then name the congruence transformation.

Studdy Solution

STEP 1

1. Angles and segments that appear congruent are indeed congruent.
2. Congruent triangles have equal corresponding sides and angles.
3. Congruence transformations include translations, rotations, reflections, and glide reflections.

STEP 2

1. Identify congruent triangles in the quilt design.
2. Verify congruence of △SUV\triangle SUV and △S′U′V′\triangle S'U'V'.
3. Determine the congruence transformation for △SUV\triangle SUV and △S′U′V′\triangle S'U'V'.
4. Verify congruence of △QRT\triangle QRT and △Q′R′T′\triangle Q'R'T'.
5. Determine the congruence transformation for △QRT\triangle QRT and △Q′R′T′\triangle Q'R'T'.

STEP 3

Identify congruent triangles in the quilt design based on the assumption that angles and segments that appear congruent are congruent. Without specific visual information, we assume all visually congruent triangles are indeed congruent.

STEP 4

Verify congruence of △SUV\triangle SUV and △S′U′V′\triangle S'U'V' using the coordinates:
- Calculate the lengths of the sides of △SUV\triangle SUV: SU = \sqrt{(0-0)^2 + (4-0)^2} = 4 \] UV = \sqrt{(2-0)^2 + (2-0)^2} = \sqrt{8} = 2\sqrt{2} \] $ VS = \sqrt{(2-0)^2 + (2-4)^2} = \sqrt{8} = 2\sqrt{2} \]
- Calculate the lengths of the sides of △S′U′V′\triangle S'U'V': S'U' = \sqrt{(0-0)^2 + (-4-0)^2} = 4 \] U'V' = \sqrt{(-2-0)^2 + (-2-0)^2} = \sqrt{8} = 2\sqrt{2} \] $ V'S' = \sqrt{(-2-0)^2 + (-2+4)^2} = \sqrt{8} = 2\sqrt{2} \]
Since all corresponding sides are equal, △SUV≅△S′U′V′\triangle SUV \cong \triangle S'U'V'.

STEP 5

Determine the congruence transformation for △SUV\triangle SUV and △S′U′V′\triangle S'U'V':
- The transformation is a reflection over the x-axis, as the y-coordinates of the vertices are negated.

STEP 6

Verify congruence of △QRT\triangle QRT and △Q′R′T′\triangle Q'R'T' using the coordinates:
- Calculate the lengths of the sides of △QRT\triangle QRT: QR = \sqrt{(-4+4)^2 + (3+2)^2} = 5 \] RT = \sqrt{(-1+4)^2 + (-2+2)^2} = 3 \] $ TQ = \sqrt{(-1+4)^2 + (-2-3)^2} = \sqrt{34} \]
- Calculate the lengths of the sides of △Q′R′T′\triangle Q'R'T': Q'R' = \sqrt{(4-4)^2 + (3+2)^2} = 5 \] R'T' = \sqrt{(1-4)^2 + (-2+2)^2} = 3 \] $ T'Q' = \sqrt{(1-4)^2 + (-2-3)^2} = \sqrt{34} \]
Since all corresponding sides are equal, △QRT≅△Q′R′T′\triangle QRT \cong \triangle Q'R'T'.

STEP 7

Determine the congruence transformation for △QRT\triangle QRT and △Q′R′T′\triangle Q'R'T':
- The transformation is a reflection over the y-axis, as the x-coordinates of the vertices are negated.

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