Topicmda64acba6167379f_1528449000663_0Topic

MidsegmentmidsegmentMidsegment

Levelmda64acba6167379f_1528449084556_0Level

Third

Core curriculummda64acba6167379f_1528449076687_0Core curriculum

IX. Analytical geometry on Cartesian plane. The student:

3) calculates the distance between two points in the coordinate system.

Timingmda64acba6167379f_1528449068082_0Timing

45 minutes

General objectivemda64acba6167379f_1528449523725_0General objective

Interpretation and the use of information presented both in a mathematical and popular science texts also using graphs, diagrams and tables.

Specific objectivesmda64acba6167379f_1528449552113_0Specific objectives

1. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

2. Finding the midsegmentmidsegmentmidsegment marked in the coordinate system and equations of a perpendicular bisectorperpendicular bisectorperpendicular bisector.

3. Applying the property of a medianmedianmedian of a triangle.

Learning outcomesmda64acba6167379f_1528450430307_0Learning outcomes

The student:

- finds the midsegmentmidsegmentmidsegment marked in the coordinate system and equations of a perpendicular bisectorperpendicular bisectorperpendicular bisector,

- applies the property of a medianmedianmedian of a triangle.

Methodsmda64acba6167379f_1528449534267_0Methods

1. Incomplete sentences.

2. Situational analysis.

Forms of workmda64acba6167379f_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmda64acba6167379f_1528450127855_0Introduction

Working in small groups, the students use the incomplete sentences technique to put their information about a segmentsegmentsegment in the coordinate system and the axial symmetry in order.

The sentences that the students should complete:

- The perpendicular bisectorperpendicular bisectorperpendicular bisector is…

- The length of segment AB, where $A(x_A;y_A),B(x_B;y_B)$ can be calculated using formula ……

- The length of segment AB, where $A(x_A;0),B(x_B;0)$ can be calculated using formula ……

- The length of segmentsegmentsegment AB, where $A(0;y_A),B(0;y_B)$ can be calculated using formula ……

The teacher verifies the answers and explains the doubts.

Proceduremda64acba6167379f_1528446435040_0Procedure

The teacher informs the students that the aim of the class is finding the midsegmentmidsegmentmidsegment marked in the coordinate system and equations of a perpendicular bisectorperpendicular bisectorperpendicular bisector.

Task
Working individually, the students analyse the material shown in the Slideshow. They formulate hypotheses, check them and formulate their conclusions.

[Slideshow]

Conclusion:
The midsegment of AB, where $A(x_A;y_A),B(x_B;y_B)$ is point $S\left(\frac{x_A+x_B}{2};\frac{y_A+y_B}{2}\right)$.mda64acba6167379f_1527752263647_0The midsegment of AB, where $A(x_A;y_A),B(x_B;y_B)$ is point $S\left(\frac{x_A+x_B}{2};\frac{y_A+y_B}{2}\right)$.

The students solve the tasks individually using the information.

Task
Find the coordinates of the midsegmentmidsegmentmidsegment of AB, if:
a) A ( 0; 7), B ( - 4; 0)

b) A ( - 6; 7), B ( - 4; 0)

c) A ( 5; 7), B ( - 4; - 7)

d) A( - 4; - 5), B ( 6; 3)

Answer:
a) S ( - 2; 3,5)

b) S ( - 5; 3,5)

c) S ( 0,5; 0)

d) S ( 1; - 1)

Task
Three vertices of parallelogram ABCD, A ( - 3; - 1), B ( 3; - 2), C ( 5; 2) and the intersection of diagonals point P ( 1; 0,5) are given. Find the coordinates of vertex D in this parallelogram. Draw parallelogram ABCD in the coordinate system.

Answer:
D ( - 1; 3)

Task
SegmentsegmentSegment AB, where A ( 6a; 4), B ( 2a; 4b) and point Sa;b, which is the midsegmentmidsegmentmidsegment of AB are given. Calculate a and b.

Answer:
$a=-\frac{2}{7},b=-\frac{8}{7}$

Discussion
How can you find the equation of the perpendicular bisectorperpendicular bisectorperpendicular bisector of segmentsegmentsegment AB? The students formulate hypotheses, check them and formulate their conclusions.

Conclusion:
In order to find the equation of the perpendicular bisector of segment AB you need to:
a) Find the coordinates of the midsegment of AB,
b) Find the equation of a line passing through points A and B,
c) Find the equation of a line perpendicular to segment AB and passing through its midsegment.mda64acba6167379f_1527752256679_0In order to find the equation of the perpendicular bisector of segment AB you need to:
a) Find the coordinates of the midsegment of AB,
b) Find the equation of a line passing through points A and B,
c) Find the equation of a line perpendicular to segment AB and passing through its midsegment.

The students solve the tasks in pairs.

Task
Find the equation of the perpendicular bisectorperpendicular bisectorperpendicular bisector of segmentsegmentsegment AB, where A ( - 2; - 10), B ( 4; 8).

Answer:
y = 3x + 4

Discussion
How to find the equation of a perpendicular bisector? The students formulate hypotheses, check them and formulate their conclusions.

Conclusion:
To find the equation of a perpendicular bisectorperpendicular bisectorperpendicular bisector you need to calculate the coordinates of the bisector of a side of a triangle, then, find the equation of a line passing through the found bisector of a side and a vertex of a triangle, which is not the end of this side.

Using the information, the students solve the tasks individually.

Task
Triangle ABC, where A ( - 1; - 4), B ( 10; 4) and C ( 4; 6) is given. Find the equation of the medians of this triangle.

Answer:
$y=-12x+54,y=\frac{9}{8}x-\frac{23}{8},y=\frac{6}{17}x+\frac{8}{17}$

Having finished all the tasks, the students present their results. The teacher assesses their work and explains any doubts.

An extra task
Calculate the area of rhombus ABCD with vertices A ( - 2; - 4), B ( 2; 6), whose diagonals intersect at point ( 2; 4).

Answer:
80

Lesson summarymda64acba6167379f_1528450119332_0Lesson summary

The students do the consolidation tasks.

Then, they formulate the conclusions to memorize:

- The midsegment of AB, where $A(x_A;y_A),B(x_B;y_B)$ is point $S\left(\frac{x_A+x_B}{2};\frac{y_A+y_B}{2}\right)$.
- In order to find the equation of the perpendicular bisector of segment AB you need to:
a) Find the coordinates of the midsegment of AB,
b) Find the equation of a line passing through points A and B,
c) Find the equation of a line perpendicular to segment AB and passing through its midsegment.
- To find the equation of a perpendicular bisector you need to calculate the coordinates of the bisector of a side of a triangle, then, find the equation of a line passing through the found bisector of a side and a vertex of a triangle, which is not the end of this side.mda64acba6167379f_1527712094602_0- The midsegment of AB, where $A(x_A;y_A),B(x_B;y_B)$ is point $S\left(\frac{x_A+x_B}{2};\frac{y_A+y_B}{2}\right)$.
- In order to find the equation of the perpendicular bisector of segment AB you need to:
a) Find the coordinates of the midsegment of AB,
b) Find the equation of a line passing through points A and B,
c) Find the equation of a line perpendicular to segment AB and passing through its midsegment.
- To find the equation of a perpendicular bisector you need to calculate the coordinates of the bisector of a side of a triangle, then, find the equation of a line passing through the found bisector of a side and a vertex of a triangle, which is not the end of this side.

Selected words and expressions used in the lesson plan

area of a rhombusarea of a rhombusarea of a rhombus

circle circumscribed on a trianglecircle circumscribed on a trianglecircle circumscribed on a triangle

coordinates of a pointcoordinates of a pointcoordinates of a point

length of a segmentlength of a segmentlength of a segment

medianmedianmedian

midsegmentmidsegmentmidsegment

perpendicular bisectorperpendicular bisectorperpendicular bisector

segmentsegmentsegment