Geometry Geometric Constructions Test Solutions and Scoring (PDF)

Geometry math test solutions: Geometric Constructions – Maths PDF

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This constructions test answers for Geometry covers geometric constructions (Common Core G-CO.D.12, G-CO.D.13), with a free PDF to print.

Use this answer key and scoring guide for the constructions 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.D.12, G-CO.D.13

Constructions test answers: worked solutions

Question 1 – Perpendicular bisector [3 pts]

AM = 10, AP = 13.

If the radius is half of AB or less, the arcs from A and B do not cross at two points.
\(AM = 10\), so \(PM = \sqrt{13^{2} – 10^{2}} = \sqrt{69}\).
\(3x + 5 = 5x – 9\), so \(x = 7\).

Scoring: 1 point per part.

Question 2 – Angle bisector [2 pts]

Each half is 56°.

\(5x + 6 = 7x – 14\)

\(x = 10\)

\(5(10) + 6 = 56^{\circ}\), so \(m\angle PVQ = 112^{\circ}\).

x = 10; 112°

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

Question 3 – Equilateral triangle [2 pts]

Both arcs have radius AB, so AC = AB and BC = AB. All three sides are equal.

Height: \(\sqrt{9^{2} – 4.5^{2}} = \sqrt{60.75} = \frac{9\sqrt{3}}{2}\).

Height 9√3/2, about 7.8

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

Question 4 – Hexagon [2 pts]

Side = radius = 10, so perimeter \(6 \cdot 10 = 60\).

The longest diagonal passes through the center: \(2 \cdot 10 = 20\).

Perimeter 60; diagonal 20

Scoring: 1 point each.

Question 5 – Square [2 pts]

Diagonal = 14.

The diagonal is 14, so \(s = \frac{14}{\sqrt{2}} = 7\sqrt{2}\).

Area \((7\sqrt{2})^{2} = 98\).

Side 7√2; area 98

Scoring: 1 point for the side, 1 point for the area.

Question 6 – Order the steps [2 pts]

The order is (c), (b), (a).

Scoring: 2 points for the correct order, 1 point if two steps are swapped.

Question 7 – Parallel line [2 pts]

Draw a transversal through P that meets l. Copy the angle that the transversal makes with l at P so that the two angles are corresponding.

Congruent corresponding angles prove the lines are parallel.

Scoring: 1 point for the method, 1 point for corresponding angles.

Question 8 – Coordinates [2 pts]

Midpoint \((1, 5)\). Slope \(\frac{7 – 3}{4 – (-2)} = \frac{2}{3}\), so the perpendicular slope is \(-\frac{3}{2}\).

\(y – 5 = -\frac{3}{2}(x – 1)\), so \(y = -\frac{3}{2}x + \frac{13}{2}\).

y = −3/2 x + 13/2

Scoring: 1 point for the midpoint and slope, 1 point for the equation.

Question 9 – Fountain [3 pts]

The fountain lies on the perpendicular bisector of AB.

It is 50 yards from B as well.

Distance from the midpoint: \(\sqrt{50^{2} – 30^{2}} = \sqrt{1600} = 40\) yards.

On the perpendicular bisector; 50 yd from B; 40 yd from the midpoint

Scoring: 1 point for the bisector, 1 point for the distance from B, 1 point for the 40 yards.

Total score

Total

20 points

Key facts behind this constructions test answers

Key facts

Constructions use only a compass and a straightedge.
Perpendicular bisector: equal arcs from both endpoints with radius more than half the segment.
Angle bisector: arc at the vertex, then equal arcs from the two crossing points.
Equilateral triangle: arcs of radius AB from A and from B.
A regular hexagon has side length equal to the radius of its circumscribed circle.

How to use this constructions 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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