# Construction Techniques 3: Perpendicular Lines and Angle Bisectors

ID: ravus-jakit
Illustrative Mathematics
Subject: Geometry

9 questions

# Construction Techniques 3: Perpendicular Lines and Angle Bisectors

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##### Problem 1

1) This diagram is a straightedge and compass construction of a line perpendicular to line ﻿$AB$﻿ passing through point ﻿$C$﻿. Explain why it was helpful to construct points ﻿$D$﻿ and ﻿$A$﻿ to be the same distance from ﻿$C$﻿.

##### Problem 2

2) This diagram is a straightedge and compass construction. Select all true statements. Write each corresponding letter in the answer box and separate letters with commas.

a) Line ﻿$EF$﻿ is the bisector of angle ﻿$BAC$﻿. b) Line ﻿$EF$﻿ is the perpendicular bisector of segment ﻿$BA$﻿.

c) Line ﻿$EF$﻿ is the perpendicular bisector of segment ﻿$AC$﻿. d) Line ﻿$EF$﻿ is the perpendicular bisector of segment ﻿$BD$﻿.

e) Line ﻿$EF$﻿ is parallel to line ﻿$CD$﻿.

##### Problem 3

3) This diagram is a straightedge and compass construction. ﻿$A$﻿ is the center of one circle, and ﻿$B$﻿ is the center of the other. A rhombus is a quadrilateral with 4 congruent sides. Explain why quadrilateral ﻿$ACBD$﻿ is a rhombus.

##### Problem 4

4) ﻿$A$﻿, ﻿$B$﻿, and ﻿$C$﻿ are the centers of the three circles. Which line segment is congruent to ﻿$HF$﻿?

a) $AB \text{ }$b) $CD \text{ }$c) $DF \text{ }$d) $CB$
##### Problem 5

5) In the construction, ﻿$A$﻿ is the center of one circle, and ﻿$B$﻿ is the center of the other. Explain why segment ﻿$EA$﻿ is the same length as segment ﻿$BC$﻿.

##### Problem 6

In this diagram, line segment ﻿$CD$﻿ is the perpendicular bisector of line segment ﻿$AB$﻿. Assume the conjecture that the set of points equidistant from ﻿$A$﻿ and ﻿$B$﻿ is the perpendicular bisector of ﻿$AB$﻿ is true.

6) Is point ﻿$M$﻿ closer to point ﻿$A$﻿, closer to point ﻿$B$﻿, or the same distance from both points?

7) Explain how you know.

##### Problem 7

8) A sheet of paper with points ﻿$A$﻿ and ﻿$B$﻿ is folded so that ﻿$A$﻿ and ﻿$B$﻿ match up with each other.

Explain why the crease in the sheet of paper is the perpendicular bisector of segment ﻿$AB$﻿. (Assume the conjecture that the set of points equidistant from ﻿$A$﻿ and ﻿$B$﻿ is the perpendicular bisector of segment ﻿$AB$﻿ is true.)

##### Problem 8

9) Here is a diagram of a straightedge and compass construction. ﻿$C$﻿ is the center of one circle, and ﻿$B$﻿ is the center of the other. Explain why the length of segment ﻿$CB$﻿ is the same as the length of segment ﻿$CD$﻿.