4.10 Square by Adjacent Midpoints

Instruction: Construct a square, given two midpoints of adjacent sides.

Goal: 7L 10E

Available tools:

• Move
• Point
• Line
• Circle
• Perpendicular Bisector
• Perpendicular
• Angle Bisector
• Intersect

Pack: Delta

Previous level: 4.9

Next level: 4.11

7L solution

Let A and B be the given midpoints.

1. Construct line AB
2. Construct the perpendicular to AB from A
3. Construct the angle bisector of ∠BAC with C an arbitrary point on the perpendicular
4. Construct the perpendicular to that angle bisector through B, intersecting AC at D
5. Construct the circle with center B and radius BD, intersecting BD at E and AC at F
6. Construct the perpendicular bisector of BE
7. Construct the perpendicular to the angle bisector of ∠BAC through F

10E solution

Let A and B be the given midpoints.

1. Construct line AB
2. Construct the perpendicular bisector of AB, intersecting AB at C
3. Construct the circle with center C and radius AC, intersecting the perpendicular bisector of AB at D and E
4. Construct line AD
5. Construct line BD
6. Construct the circle with center E and radius DE, intersecting AD at F, DE at G and BD at H
7. Construct line FG
8. Construct line GH

2V solution

From the 7L solution: Let G be the intersection of the perpendicular bisector of BE and AB

1. Construct the perpendicular to the perpendicular bisector from G
2. Construct the line BF
3. Construct the perpendicular to BF through A
4. Construct the perpendicular to the previous perpendicular through D

From the 10E solution: Let I be the intersection of FG and AB, and J the intersection of GH and AB

1. Construct the perpendicular to FI from I
2. Construct the perpendicular to HJ from J
3. Construct line AE
4. Construct line BE