Geometry Symmetry of Polygons and Rigid Motions Test Solutions and Scoring (PDF)

Geometry math test solutions: Symmetry of Polygons and Rigid Motions – Maths PDF

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This geometry symmetry test answers for Geometry covers symmetry of polygons and rigid motions (Common Core G-CO.A.3, G-CO.B.6), with a free PDF to print.

Use this answer key and scoring guide for the symmetry test. Each solution shows the full work, and the scoring note tells how the points are split between method and final answer. Partial credit is listed so that grading stays consistent from one paper to the next.

Geometry · Common Core State Standards: G-CO.A.3, G-CO.B.6

Geometry symmetry test answers: worked solutions

Question 1 – Nonagon symmetry [2 pts]

Regular nonagon with its 9 lines of symmetry.

9 lines of symmetry.

\(360^{\circ} \div 9 = 40^{\circ}\)

9 lines; 40°

Scoring: 1 point for each correct value.

Question 2 – Rectangle symmetries [2 pts]

Rotation of 180° about the center.

Reflection across the vertical line through the midpoints of AB and CD.

Reflection across the horizontal line through the midpoints of BC and DA.

Scoring: 1 point for the rotation, 1 point for both reflections.

Question 3 – Isosceles triangle [3 pts]

Dashed line x = 3.

\(PR = \sqrt{2^{2} + 3^{2}} = \sqrt{13}\) and \(QR = \sqrt{(-2)^{2} + 3^{2}} = \sqrt{13}\)
The reflection across \(x = 3\) is \((x, y) \text{ maps to } (6 – x, y)\). \(P(1,1) \text{ maps to } (5,1) = Q\), \(Q \text{ maps to } P\), \(R(3,4) \text{ maps to } (3,4) = R\). Vertices map to vertices.

(a) PR = QR = √13; (b) P↔Q and R stays fixed

Scoring: (a) 1 point. (b) 1 point for the rule, 1 point for the three images.

Question 4 – Rotation without reflection [2 pts]

A 180° rotation about O, the intersection of the diagonals, sends A to C and B to D.

A reflection across a diagonal would send a vertex to a point that is not a vertex, because the diagonals are not perpendicular bisectors of the sides.

Scoring: 1 point for the rotation, 1 point for the explanation.

Question 5 – Preserved measures [2 pts]

Triangle DEF.

DE = 9, EF = 12 and m∠E = 90° because rotations preserve distance and angle measure.

DF = \(\sqrt{9^{2} + 12^{2}} = 15\).

DE = 9, EF = 12, m∠E = 90°, DF = 15

Scoring: 1 point for the preserved values, 1 point for DF.

Question 6 – 15-gon [3 pts]

15 lines of symmetry.

\(360^{\circ} \div 15 = 24^{\circ}\)

The rotations are multiples of 24° up to 360°, so there are 15.

15 lines; 24°; 15 rotations

Scoring: 1 point for each correct value.

Question 7 – Square vertices [2 pts]

A maps to B, B to C, C to D, D to A.

\((4,0) \text{ maps to } (0,4),\ (0,4) \text{ maps to } (-4,0),\ (-4,0) \text{ maps to } (0,-4),\ (0,-4) \text{ maps to } (4,0)\).

Every vertex maps to a vertex, so this 90° rotation is a symmetry.

Scoring: 1 point for the images, 1 point for the conclusion.

Question 8 – Letters [2 pts]

H, N, S and Z all look the same after a 180° turn.

Only H has a horizontal line of symmetry (and a vertical one).

180° rotation: H, N, S, Z; horizontal line: H

Scoring: 1 point for each part.

Question 9 – Fan blades [2 pts]

\(360^{\circ} \div 5 = 72^{\circ}\).

Rotations: 72°, 144°, 216° and 288°.

72° apart; 72°, 144°, 216°, 288°

Scoring: 1 point for the angle, 1 point for the list.

Total score

Total

20 points

Key facts behind this geometry symmetry test answers

Key facts

A regular n-gon has n lines of symmetry and rotational symmetry every 360/n degrees.
A rectangle that is not a square has 2 lines of symmetry and a 180 degree rotation.
A parallelogram that is not a rectangle or rhombus has no line of symmetry but has a 180 degree rotation.
Rigid motions preserve distance and angle measure.
To test a symmetry, check that every vertex maps to a vertex.

How to use this geometry symmetry test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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