Geometric Constructions: Geometry Math Test – PDF Download

Question 1: Perpendicular bisector measurements (2 points)

Arcs of radius 13 cm centered at \(A\) and \(B\) cross at \(P\) and \(Q\), where \(AB=10\text{ cm}\). Line \(PQ\) meets \(AB\) at \(M\).

1. Find \(AM\). (1 point)
2. Find \(PQ\). (1 point)

Question 2: Angle bisector with a variable (2 points)

Ray \(\vec{BD}\) bisects \(\angle ABC\), with \(m\angle ABD=(4x-6)^{\circ}\) and \(m\angle DBC=(3x+9)^{\circ}\).

Find \(x\) (1 point) and \(m\angle ABC\) (1 point).

Question 3: Perpendicular from a point (3 points)

Point \(P\) is not on line \(l\). Describe, in order, the steps to construct the line through \(P\) perpendicular to \(l\) with a compass and straightedge.

Question 4: Justify an angle copy (2 points)

In the construction, \(AP=AQ=DP^{\prime}=DQ^{\prime}\) and the compass width \(PQ\) was copied to \(P^{\prime}Q^{\prime}\). Explain why \(\angle QAP\) is congruent to \(\angle Q^{\prime}DP^{\prime}\).

Question 5: Parallel line construction (2 points)

A student constructs a line through \(P\) parallel to line \(l\) by copying the \(58^{\circ}\) angle at \(Q\) to point \(P\).

1. What is the measure of the copied angle at \(P\)? (1 point)
2. Which theorem shows that the new line is parallel to \(l\)? (1 point)

Question 6: Equilateral triangle (2 points)

Triangle \(ABC\) is constructed on \(\overline{AB}\) with \(AB=9\text{ in}\) by drawing arcs of radius \(AB\) from \(A\) and \(B\).

1. Find the perimeter of triangle \(ABC\). (1 point)
2. Find \(m\angle C\) and give a reason. (1 point)

Question 7: Regular hexagon in a circle (3 points)

A regular hexagon is inscribed in a circle of radius 8 cm using the compass-and-straightedge method.

1. Find the length of each side. (1 point)
2. Find the perimeter. (1 point)
3. Find the measure of each interior angle of the hexagon. (1 point)

Question 8: Square in a circle (2 points)

Square \(ABCD\) is inscribed in a circle with center \(O\) and diameter 12 in.

1. Find \(AC\). (1 point)
2. Find \(m\angle AOB\). (1 point)

Question 9: Compass width (2 points)

Maya wants to bisect \(\overline{AB}\), where \(AB=8\text{ cm}\). She sets the compass to 3 cm and draws arcs centered at \(A\) and \(B\). Explain what goes wrong, and give the condition that the compass radius must meet.

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