Geometry

1.1 Defining Terms

1.1 Defining Terms

Point

A point is a location in space.

A point is denoted with a capital letter.

Line

A straight line is an infinite number of points which extends in two opposite directions indefinitely.

A line is denote with a lower case letter or two capital letters with a double headed arrow above the two letters.

Triangle

A triangle is a figure that is formed by joining three non-collinear points. The sum of the measures of the angles of a triangle is 180^o.

Ray

A ray is half a line.

A ray is denoted as two capital letters with a single headed arrow above the two letters. The ray is written with the terminal point first.

Angle

An angle is formed by the intersection of two rays with one common endpoint. This endpoint is called the vertex.

Segment

A segment is a finite portion of a line.

A segment is denoted as two capitol letters with a line above the two letters.

Plane

A plane is flat surface of space.

A plane is denoted by three capital letters or with one lower case letter.

Collinear

Three or more points are collinear if they all lie on the same line.

Coplanar

Four or more points are coplanar if they lie on the same plane.

The Distance Formula

The distance, d, between two points, (x₁, y₁), (x₂, y₂), on a coordinate plane is given by:

Example 1:

Find the distance between the following points: (-4, -3), (8, -2).

Solution:

Therefore the distance is

Student #1:
First, find the intersection of the lines y = 1 and y = -x.
Second, use the above point and the distance formula to find the distance between the intersection point and the point (2,5).

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Answer

Bisectors:

Midpoint

The midpoint, or bisector of a segment, is the middle of a segment.

The Midpoint Formula

The midpoint M of two points, (x₁, y₁), (x₂, y₂), on a coordinate plane is given by:

Example 2:

Find the midpoint of the segment between (-4,2)(-8,-10).

Solution:

Therefore the coordinate point of the midpoint is (-6,-4).

Student #2:
Find the midpoint of the line y = 2x on the interval -2< x < 2.

(0,0)

(2,4)

(-1,2)

(-2,1)

Answer

Mid-Ray

The mid-ray or bisector of an angle divides the angle in half.

Constructions:

It is possible to construct many geometric figures without making a measurement. Before the invention of rulers or other measuring devices, Euclid used a straightedge and a compass to construct geometric figures. This is known as construction.

Copy a Segment:

Example 3:

Make an exact copy of the following segment.

Solution:

1.    To construct a copy of a segment, take a straight edge and draw a line obviously longer than the original segment.

2.    Place the compass on point A of the original diagram and extend it to point B.

The compass now contains the measurement of the segment. Do not let the compass change settings.

3.    Place the compass at point A^/ on the copied line diagram, and mark the point B^/.

An exact copy of the original segment is complete.

_____________________

Bisect a Segment:

Example 4:

Find the exact midpoint of a segment by bisecting the following segment.

Solution:

1.    Place the compass on point A and open the compass to a location which is more than half way on the segment. Draw a semicircle at this location. Do not let the compass change settings.

2.    Place the compass on point B and with the same settings as with point A, draw a semicircle at this location.

Mark the two intersection points of the semicircles, and draw a line connecting the points.

The point where the two lines intersect is the midpoint of the segment.

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Copy an Angle:

Example 5:

Make an exact copy of the following angle:

Solution:

1.    Draw a horizontal ray

2.    Place the compass on point O and draw a semicircle. Do not let the compass change settings.

3.    Place the compass on O^/ and draw a semicircle.

4.    Mark the intersection points as X and Y on the original diagram and Y^/ on the intersection of the copied diagram.

5.    Measure the distance from X to Y on the original diagram.

6.    Without changing the compass settings, place the compass on point Y^/ on the copied diagram and make a mark on the semicircle. Label this point X^/.

7.    Draw ray through X^/.

An exact copy of the original angle is complete.

_____________

Bisect an Angle:

Example 6:

Find the exact midpoint of an angle by bisecting the following angle.

Solution:

1.    Place the compass on point O, and then make an arc across through the two rays of the angle. Mark the intersection points as X and Y.

2.    Without changing the compass settings, place the compass on point X, and make a mark out in the center of the angle.

3.    Without changing the compass settings, place the compass on point Y, and make a mark out in the center of the angle.

4.    Label the intersection point of these two marks as point R.

5.    Draw ray .

The ray which intersects the angle, is the mid-ray of the angle.

Student #3:
Copy the angle and then bisect

Click answer to view the solution.

Answer

Student #4:

A. Copy
B. Bisect    (Label this ray )

C. Draw a segment from point A to point B.

D. Label the intersection of and as Q.

E. Find the midpoint of the line by bisecting the segment and label this point M.

Answer

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Student #1

Answer is C:

Solution:

The intersection point of the two lines is (-1,1).

The distance between (-1,1) and (2,5) is given by:

Therefore the distance is 5 units.

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Student #2:

Answer is A:

Solution:

The midpoint of (2,4),(-2,-4) is found as follows:

Therefore the midpoint is (0,0).

Go Back to Lesson

Student #3:

Solution:

Step A: Copy the angle:

1.    Draw a horizontal ray

2.    Place the compass on point O and draw a semicircle. Do not let the compass change settings.

3.    Place the compass on O^/ and draw a semicircle.

4.    Mark the intersection points as X and Y on the original diagram and Y^/ on the intersection of the copied diagram.

5.    Measure the distance from Y to X on the original diagram.

6.    Without changing the compass settings, place the compass on point Y^/ on the copied diagram and make a mark on the semicircle. Label this point X^/.

7.    Draw ray through X^/.

Thus the angle is copied.

Step B: Bisect the original angle.

1.    Place the compass on point O, and then make an arc across through the two rays of the angle. Mark the intersection points as X and Y.

2.    Without changing the compass settings, place the compass on point Y, and make a mark out in the center of the angle.

3.    Without changing the compass settings, place the compass on point X, and make a mark out in the center of the angle.

4.    Label the intersection point of these two marks as point R.
5.    Draw ray .

The ray which intersects the angle, is the mid-ray of the angle.

The construction is complete.

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Student #4:

Solution:

Step A: Copy the angle:

1.    Draw a horizontal ray

2.    Place the compass on point O and draw a semicircle. Do not let the compass change settings.

3.    Place the compass on O^/ and draw a semicircle.

4.    Mark the intersection points as X and Y on the original diagram and Y^/ on the intersection of the copied diagram.

5.    Measure the distance from Y to X on the original diagram.

6.    Without changing the compass settings, place the compass on point Y^/ on the copied diagram and make a mark on the semicircle. Label this point X^/.

7.    Draw ray through X^/.

Thus the angle is copied.

Step B: Bisect the original angle.

1.    Place the compass on point O, and then make an arc across through the two rays of the angle. Mark the intersection points as X and Y.

2.    Without changing the compass settings, place the compass on point X, and make a mark out in the center of the angle.

3.    Without changing the compass settings, place the compass on point Y, and make a mark out in the center of the angle.

4.    Label the intersection point of these two marks as point R.
5.    Draw ray .

The ray which intersects the angle, is the mid-ray of the angle.

Step C: Draw a segment from point A to point B.
Step D:

Step E:

The pink point shows the midpoint of the segment.

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