Geometry: Points, Lines, Planes, and Angles

MA.912.G.1.1 Find the lengths and midpoints of line segments in two-dimensional coordinate systems. Block 11

Length of a segment Length of a segment if a distance between its endpoints We will start to review the distance between two points first by finding the distance on the number line The distance between the points on the number line is the absolute value of the difference of their coordinates

Number line Distance: E to A is |4-(-4)|=|8|=8 What are the distances: AB, AC, AD, BC, BD, BE? You can use manipulative: http://www.allmathwords.org/en/a/absolutevalue_cd.htmlfor classroom demonstration of absolute value. Then open the GeoGebra files: number_line.ggb, number_line_addition.ggb, number_line_distanec.ggb for the demonstration and ask students to answer the question number 1 in the handout.

Cartesian coordinate : position of a point Coordinates of a point A=(x,y) on the coordinates plane is determined by two numbers x and y which corresponds to the position of points Ax on x-axis and and Ay on the y-axis. Explain the relation of position of point A and its coordinates on the plane. Make students open GeoGebra file cartesian_coordinates.ggb manipulate the point A and make observation about the coordinates of A. Then make students open the cartesian_coordinates_test.ggb to practice finding points with given coordinates on the coordinate plane for review. Next ask them to complete Task 1 from the handout.

Distance between points on the coordinate plane

Pythagorean theorem Pythagorean theorem is a basis to find a distance between points on the coordinate plane First we review the Pythagorean theorem: for the right triangle the following holds: a2+b2=c2 Open files for illustration of Pythagorean theorem: Pythagoras_proof_1.ggb and Pythagorar_proof_2.ggb and open the web page: http://standards.nctm.org/document/eexamples/chap6/6.5/index.htm for illustration of Pythagorean theorem

Distance formula and Pythagorean theorem Distance formula can be derived form Pythagorean theorem This could be illustrated by the following example:

Pythagorean theorem vs distance formula Coordinates of points: A B, and C are: Open the file: distance_cartesian_coordinates.ggb to illustrate the concept

Pythagorean theorem vs distance formula Lengths of the sides: a,b and c are: a=|x2-x1| b=|y2-y1| And c can be calculated: c2=a2+b2 Open the file: distance_cartesian_coordinates.ggb to illustrate the concept

Pythagorean theorem vs distance formula We can also state the distance formula in more common way: If A and B are points on the coordinate plane with coordinates: Now ask students to answer question Q3 from the handout. Open file: distance_cartesian_coordinates.ggb for demonstration

Midpoint of a segment Midpoint of a segment is a point halfway between the endpoints of a segment If M is a midpoint of a segment then AM=BM Open file: midpoint_on_the_number_line.ggb for the demonstration

Midpoint of a segment On the number line midpoints represents the mean or average of the coordinates of the coordinates of the endpoints The formula for coordinates of the midpoint is: Open file: midpoint_on_the_number_line.ggb for the demonstration

Midpoint of a segment Midpoint formula in the coordinate plane: Open the file midpoint_ilustration_coordinates_plane.ggb for interactive illustration of the formula. Handout question Q4 and Q5

Midpoint of a segment - construction Construction of a bisection of a segment Make student refer to the handout and follow the steps of construction in Task 2 from there to create midpoint and perpendicular bisector of a segment in three ways. You can also refer to creating_perperdicular_bisector.ggb file

Review and questions Discuss with students the different ways to construct perpendicular bisector of a segment What are the advantage and disadvantages of different methods? Class discussion, Question 6/