Q4 Project Example

Q4 Project Example

​Geometric Constructions

By Colin Zehnder

A compass is used to draw circles and intersecting arcs to create points. A compass point is placed on a point on the page to center the circle.​

Compass

​A straightedge is any tool like a ruler used to draw a straight line. These connect points made by intersecting arcs of the compass.

Straightedge

Many constructions start by connecting two points with a straight edge to make a line segment as shown above.

Segments

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Intersecting arcs are used to form new points to connect in a construction. The animation above shows a compass used to intersect arcs created at point A with arcs created at point B.​

Intersecting Arcs​

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​Tools of Construction

​To construct an angle bisector to the given angle ABC.

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1)Place the compass centered on vertex B and set to any width that fits on the angle.

2)Draw a circle centered at B that intersects the angle twice.

3)Label one intersection point as E and the other as D.

4)Set the compass to a width that will pass about half way through the middle of the angle. 5)You may set the compass to the width ED for example.

6)Center the compass at E and draw a circle.

7)Center the compass at D and draw a circle.

8)Label the intersection of the circles as point F.

9)With a straight edge draw ray BF to complete the angle bisector.

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Angle Bisector

To construct a perpendicular bisector to given segment AB:

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1)Place a compass centered on A and set to a length wider than about the middle of the segment.

2)Draw a circle centered at A.

3)Keep the width the same and draw a circle centered at B.

4)Label one point where the circles intersect as C and the other as D.

5)Connect C & D with a straightedge to complete the bisector.​

Perpendicular Bisector

​Basic Constructions

Multiple Choice

What construction is being completed in this image?

• Construction of equilateral triangle ACB.

• Construction of a perpendicular bisector CD to segment AB.

• Construction of the right angle CAD.

• Construction of a congruent segment CD to segment AB.

Multiple Select

What is the next step in constructing the angle bisector of angle PQR?

• Draw a ray from point Q to the intersecting arcs.

• Set the compass wide enough to draw arcs intersecting in the middle of the angle.

• Label the intersecting arcs in the middle of the angle with a point.

• Construct a circle centered at point Q to the width of the intersecting arcs.

Multiple Select

What is this?

• A phrog

• The history of the entire world, I guess

• 100% a frog

• Potion Seller

To construct a line parallel to given segment AB and a point C not on line AB:​

1)Draw a line passing and continuing through point C and any point in the interior of segment AB. Label the intersection point D.

2)Place the compass centered at point D​ and set to a width approximately at the midpoint of CD. Draw a circle centered at D.

​3)Label the intersection of the circle and CD as point E and the intersection of the circle and DB as F.

4)Using the same compass with, draw a circle through line CD above point C and label the intersection point G.​

5)Place the compass on point E and set the width to EF.

6)Move the compass to center G and draw a circle. Label new intersection point between circles as point H.

7)Draw a line between CH. Line CH is parallel to AB and line CD is the transversal.​

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Construct a Parallel Line and Transversal

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Transversals and Angles Pairs

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​When two parallel lines are intersecting by a transversal, 8 angles are formed and there are multiple congruent angle pairs.

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Corresponding Angle Pairs

1 & 5, 3 & 7, 2 & 6, and 4 & 8​

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Alternate Interior Angle Pairs

3 & 6 and 4 & 5

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Alternate Exterior Angle Pairs​

1 & 8 and 7 & 2​

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Vertical Angle Pairs

1 & 4, 2 & 3, 5 & 8, and 6 & 7​

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Multiple Choice

What is the least information necessary to start constructing a line parallel to a line segment.

• A line segment.

• A line segment and its transversal.

• A line segment and its midpoint.

• A line segment and a parallel line segment.

Multiple Choice

Which theorem proves the angles labelled with beta (β) congruent?

• Corresponding Angles Theorem

• Alternate Exterior Angles Theorem

• Alternate Interior Angles Theorem

• Vertical Angles Theorem

Multiple Select

Who is this?

• Amogus

• Ben the Talking Dog

• Towelie

• My Sleep Paralysis Demon

The altitude of a triangle is a segment that is considered the height of the triangle. The altitude forms a right angle with a leg of a triangle and passes through the vertex opposite the leg.​​

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To construct an altitude as in the example, create a circle centered at A with radius AB. Construct a circle centered at C with radius CB. Connect angle B with a segment to the intersection of both circles.​

​​Altitude of a Triangle

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​When three altitudes are constructed, the altitudes intersect at a point known as the orthocenter. This point is the center of gravity of the triangle! If you cut out the triangle and found the orthocenter, it would balance on a pencil best when the pencil meets the orthocenter.

​Orthocenter​

​Advanced Constructions (Part 1)

The median of a triangle is found by connecting a vertex to the midpoint of a leg opposite of the vertex.

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To construct a median:

1)Draw a circle with center A and radius AC

2​)Draw a circle with center C and radius CA

3)Draw a segment between the intersection of both circles.

4)Label the intersection of the segment and AC as point D.

5)Draw a segment from point D to vertex B.​

Median of a Triangle

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Three medians meet at a point known as the centroid. In physics, the centroid of an object is its center of gravity. If you cut out this triangle, it would balance on a pencil best when the pencil point is on the centroid.

The centroid is also considered the geometric center of an object. When trying to capture a criminal, you might find their lair by finding the centroid between three of their crime scenes.

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Centroid

​Advanced Constructions (Part 2)

Multiple Choice

Which construction would you use if you were trying to find the area of a triangle using 1/2(base)(height)?

• Altitude

• Median

• Perpendicular Bisector

• Angle Bisector

Multiple Choice

Sadmep was trying to construct a median from vertex A to side BC. Sadmep realized they made a mistake in their construction? What was the mistake and how could it be fixed?

• E is connected to A with a segment. E and D should connect to find the midpoint of BC.

• The circles are centered at B and C. Circles should be centered at A instead.

• E is connected to A. A and D should connect with a line until intersecting BC.

• The circles are centered at B and C but with the wrong radii. Use radii BA & CA instead.

Draw

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