Performing Constructions Involving Triangles

The steps to find midpoint C of AB by constructing a perpendicular bisector using a straight edge and compass are listed below. Select the steps in order from top to bottom.

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The steps to construct equilateral ΔTUV using line m are given below. Place the steps in order from top to bottom.

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Which of these are correct statements about the process for constructing an angle bisector? Select all that apply.

• Any two points on the rays of the angle can be used to determine an angle bisector.

• The same radius must be used when finding the intersection of the two arcs drawn from a point on each of the rays of the angle.

• From the vertex point, the same radius must be used to determine the two points on the rays of the angle.

• Any set of radii can be used to construct an angle bisector.

• A point that appears to be halfway between the two rays of the angle can be used to draw the angle bisector.

Karl constructed two line segments AB and PQ, and then he drew line segments to connect AP and PB completing △APB. Karl claims that PQ is the perpendicular bisector of AB. His friend Ava took some measurements to verify Karl's claim.

Which of Ava's diagrams contain sufficient information to guarantee that PQ is the perpendicular bisector of AB? Select all that apply.

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The construction below shows the bisection of ∠A.

Which of the following can be used to prove that ⊿ABD is congruent to ⊿ACD by the SAS triangle congruence theorem? Select all that apply.

• A. ∠BAD ≅ ∠ACB

• B. ∠BAD ≅ ∠CAD

• m∠BDA = m∠CDA = 90°

• segment AD ≅ segment AD

• segment AB ≅ segment AC

The construction below shows line DC as the perpendicular bisector of segment AB.

Which of the following statements can be used to prove ⊿ADM ≅ ⊿BDM by the SAS triangle congruence theorem? Select all that apply.

• ∠ADM ≅ ∠BDM

• ∠MAD ≅ ∠MBD

• m∠AMD = m∠BMD = 90°

• segment AM ≅ segment BM

• segment DM ≅ segment DM

Angel wanted to construct a line perpendicular to line AB through point P. To begin, she opened her compass so that it extended from P past line AB. She then marked the two arcs shown in the diagram below.

Which option below best describes the next step(s) Angel should complete in her construction?

• Use the straightedge to draw lines through P and the points where each arc intersects line AB

• Place the end of the compass on the lef-hand arc where it intersects line AB. Draw a new arc through A. Repeat this step using the righthand red arc and B.

• Place the end of the compass on the left-hand arc and extend it more than halfway to the right-hand arc. Draw an arch below line AB and directly below P. Repeat this process starting fro the right-hand arc.

• Estimate the midpoint of the segment on line AB created by the two arcs. Use the straightedge to draw a line from P to this new point.

• Place end of the compass at B and adjust the compass to create an arc through A. Repeat this process, reversing B and A.

Macy used a straightedge to draw ⊿RST. Then, she used a compass to find points U and V, equidistant from point S. Next, she used her compass to find point W, equidistant from points U and V. Finally, Macy used her straightedge to draw ray SW before labeling point X, the intersection of ray SW and side RT of ⊿RST. Her construction is shown below.

If Macy has performed these steps correctly, which of the following statements are true? Select all that apply.

• Segment RT and Ray SW are perpendicular.

• Point X is the midpoint of segment SW.

• Ray SW is the angle bisector of ∠RST.

• m∠RST = 2⋅m∠RSX

• UV = VW

Starting with angle A, triangle ABG was constructed as follows:

1) The tip of the compass was put on A, and an arc was made. Points C and E were labeled on this arc.

2) Set at the same length as Step 1, the tip of the compass was put on B, and an arc was made. Point D was labeled on this arc.

3) The tip of the compass was put on C and an arc was made through E.

$) Set at the same length as Step 3 the tip of the cmpass was put on D and an arc was made. Point F was labeled where this arc intersected with the arc from Step 2.

5) The straightedge was used to draw a ray extending from B through F. The intersection of this ray with Ray AE was labeled G.

Select the true statement(s) about the triangle created from this construction. Select all that apply.

• Angle A is congruent to angle B.

• Angle B is congruent to angle G.

• Side AG is congruent to side BG.

• Triangle ABG must be an isosceles triangle.

• Side AB is congruent to side AG.

You want to find the midpoint of line segment JK by using a perpendicular bisector. Which of the steps below is NOT correct?

• Set the compass to less than halfway between J and K.

• Draw an arc centered on point J toward point K using the set radius.

• Draw a similar arc centered on point K toward point J.

• Connect the two intersecting points of the arcs to create a perpendicular bisector.