Topicmf91f75410e019c76_1528449000663_0Topic

Similar trianglessimilar trianglesSimilar triangles. Examples

Levelmf91f75410e019c76_1528449084556_0Level

Third

Core curriculummf91f75410e019c76_1528449076687_0Core curriculum

VIII. Plane geometry. The student:

8) uses the properties of similar trianglessimilar trianglessimilar triangles;

12) conducts geometric proofs.

Timingmf91f75410e019c76_1528449068082_0Timing

45 minutes

General objectivemf91f75410e019c76_1528449523725_0General objective

Reasoning, including multiple‑stage arguments, giving arguments, justifying the correctness of reasoning, distinguishing a proof from an example.

Specific objectivesmf91f75410e019c76_1528449552113_0Specific objectives

1. Application of the similarity of triangles for solving tasks.

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

Learning outcomesmf91f75410e019c76_1528450430307_0Learning outcomes

The student:

- applies of the similarity of triangles for solving tasks,

- conducts geometric proofs applying of the similarity of triangles.

Methodsmf91f75410e019c76_1528449534267_0Methods

1. Discussion.

2. Situational analysis.

Forms of workmf91f75410e019c76_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmf91f75410e019c76_1528450127855_0Introduction

The teacher informs the students that during the class they will use the properties of similar trianglessimilar trianglessimilar triangles to solve geometry tasks, including the tasks involving proofs.

The students recollect the properties of similar trianglessimilar trianglessimilar triangles: AAA, SSS, SAS.

They also revise the concept of similarity ratiosimilarity ratiosimilarity ratio.

Working in pairs, the students decide if triangles with given chacteristics are similar.

Task
In which of the following cases triangles $ABC$ and $PQR$ are similar?

A)
$\angle A={40}^{\circ },\angle B={60}^{\circ },\angle C={80}^{\circ },\angle P={40}^{\circ },\angle Q={60}^{\circ },\angle R={80}^{\circ }$

B)
$\angle A={40}^{\circ },\angle B={70}^{\circ },\angle C={70}^{\circ },\angle P={50}^{\circ },\angle Q={65}^{\circ },\angle R={65}^{\circ }$

C)
$\left|AB\right|=2,5 cm, \left|BC\right|=4,5 cm, \left|CA\right|=3,5 cm, \left|PQ\right|=5 cm, \left|QR\right|=9 cm, \left|RP\right|=7 cm$

D)
$\left|AB\right|=3 cm, \left|BC\right|=5 cm, \left|CA\right|=5 cm, \left|PQ\right|=4,5 cm, \left|QR\right|=7,5 cm, \left|RP\right|=6 cm$

Proceduremf91f75410e019c76_1528446435040_0Procedure

The students work in pairs using their computers. They observe three examples of similar trianglessimilar trianglessimilar triangles, which are commonly used in geometry tasks.

Task
Open Geogebra applet - „Examples of similar trianglessimilar trianglessimilar triangles”.

[Geogebra applet]

Do the following tasks.
1. A line parallel to the base of triangle ABC cuts off a triangle similar to triangle ABC. Analyse the diagram in the applet. Which criterion of similarity can you use to justify that triangles ABC and DEC are similar? Move segment DE parallelly to base AB. How does the similarity ratio of triangles change? Pay special attention to case k = 2.mf91f75410e019c76_1527752263647_01. A line parallel to the base of triangle ABC cuts off a triangle similar to triangle ABC. Analyse the diagram in the applet. Which criterion of similarity can you use to justify that triangles ABC and DEC are similar? Move segment DE parallelly to base AB. How does the similarity ratio of triangles change? Pay special attention to case k = 2.

2. The triangles formed by diagonals of a trapezoid and including its bases are similar. Analyse the diagram in the applet. Which criterion of similarity can you use to justify that triangles ASB and DSC are similar? Change the position of the vertices of the trapezoid and check when the triangles including the arms of the trapezoid are similar. Change the position of the vertices so as to form a parallelogram. Which triangles are similar?mf91f75410e019c76_1527752256679_02. The triangles formed by diagonals of a trapezoid and including its bases are similar. Analyse the diagram in the applet. Which criterion of similarity can you use to justify that triangles ASB and DSC are similar? Change the position of the vertices of the trapezoid and check when the triangles including the arms of the trapezoid are similar. Change the position of the vertices so as to form a parallelogram. Which triangles are similar?

3. The altitude of a right triangle dropped from the vertex of the right angle divides the triangle into two similar triangles. Analyse the diagram in the applet. Which criterion of similarity can you use to justify that triangles ADC and CDB are similar? Are these triangles also similar to triangle ABC? Why?mf91f75410e019c76_1527712094602_03. The altitude of a right triangle dropped from the vertex of the right angle divides the triangle into two similar triangles. Analyse the diagram in the applet. Which criterion of similarity can you use to justify that triangles ADC and CDB are similar? Are these triangles also similar to triangle ABC? Why?

A selected pair of students presents the results, which are discussed with the whole class.

Working in groups the students use the observation that they made while working with applet: „Examples of similar triangles” and they prove the theorem about the midsegmentmidsegmentmidsegment in a triangle.

The theorem - about the midsegmentmidsegmentmidsegment in a triangle.

- The segment joining the midpoints of two sides of a triangle is parallel to the third side and it is twice shorter than the third side.

The groups present the proofs for the theorem. The teacher pays attention to the correctness of the proof of the parallelism of the midsegmentmidsegmentmidsegment to the base.

The students solve the tasks individually, present their solutions and explain any doubts.

Task
In triangle ABC segment DE parallel to base AB was drawn. Find the length of segment BC if |EF| = 10, |AB| = 15, |EB| = 9.

Task
In trapezoid ABCD diagonals AC and BD intersect at point S. Find the length of segment DS, if |AB| = 18, |DC| = 6, |DB| = 16.

Task
In right triangle ABC altitude CD was dropped from the vertex of the right angle. Calculate the length of leg AC, if |CD| = 4, |BD| = 3.

An extra task:
Trapezoid $ABCD$ with bases $\left|AB\right|=a$ and $\left|CD\right|=b$ is given. The line going through the intersection point of the diagonals of the trapezoid intersects the sides of the trapezoid in points $\left|MN\right|=\frac{2ab}{a+b}$.

Lesson summarymf91f75410e019c76_1528450119332_0Lesson summary

The students do the consolidation tasks.

Then, theysummarize the class and formulate the conclusions to memorize.

- A line parallel to the baseparallel to the baseparallel to the base of triangle ABC cut off a triangle similar to ABC.

- The midsegmentmidsegmentmidsegment in a triangle is parallel to its base and equals half of its length.

- The triangles formed by the diagonals of a trapezoid and including its base are similar.

- The altitude in a right triangles dropped from the vertex of the right angle divides the triangle into two similar trianglessimilar trianglessimilar triangles.

Selected words and expressions used in the lesson plan

midpointmidpointmidpoint

midsegmentmidsegmentmidsegment

parallel to the baseparallel to the baseparallel to the base

similar trianglessimilar trianglessimilar triangles

similarity ratiosimilarity ratiosimilarity ratio