RELAÇÕES MÉTRICAS NO TRIÂNGULO RETÂNGULO \Prof. Gis/

Name: RELAÇÕES MÉTRICAS NO TRIÂNGULO RETÂNGULO \Prof. Gis/
Uploaded: 2021-07-21T17:00:38.000Z
Duration: 1 h 22 min

Understanding Metric Relationships in Right Triangles

Introduction to Right Triangles

The class begins with an introduction to the topic of metric relationships in right triangles, emphasizing the definition of a right triangle as one that contains a 90° angle.

The term "metric relationships" refers to the connections between the measurements of the sides of a right triangle.

Components of a Right Triangle

A specific triangle, labeled ABC, is introduced. It is noted that this triangle has its right angle at vertex A.

The hypotenuse is identified as the longest side opposite the right angle (vertex A), and it is referred to as "azinho."

The other two sides are named: segment AB (leg facing vertex C) is called "cezinho," and segment AC (leg facing vertex B) is called "baby."

Visualizing Height and Projections

The height from vertex A to the hypotenuse is discussed, which forms a right angle with it. This height is denoted as 'h.'

Additional segments are introduced: M and N represent projections from points B and C onto the hypotenuse respectively.

Understanding Projections through Visualization

Projections M and N are explained using an analogy involving shadows created by light sources, helping students visualize these concepts better.

The instructor humorously engages with students about visualizing shadows while explaining how projections work.

Formation of Similar Triangles

By drawing height 'h,' two smaller triangles (ABH and AHC) are formed within triangle ABC.

Understanding Similar Triangles and Their Relationships

Introduction to Triangle Similarity

The discussion begins with the concept of triangles having equal internal angles, emphasizing that similar triangles share this property.

Three specific triangles are introduced, highlighting their similarity due to their interior angles, including a right angle and a pink angle.

The speaker identifies the three triangles as triangle 1, triangle 2, and triangle 3 for clarity in comparison.

Visualizing Triangle Measurements

The speaker illustrates the sides of the triangles: hypotenuse (A), side B, and side C.

Measurements are assigned to each side; side C is identified as originating from one triangle while height (h) is noted as coming from above.

A projection measurement (N) is introduced alongside the height measurement for better visualization of relationships among the triangles.

Establishing Proportional Relationships

The speaker begins comparing triangle 1 with triangle 2, noting that corresponding sides must be proportional based on their angles.

Side B in triangle 1 corresponds to side H in triangle 2 since they face the same purple angle.

Further relationships are established: side C in triangle 1 corresponds to N in triangle 2, while hypotenuse A relates to hypotenuse C.

Continuing Comparisons Among Triangles

Next, comparisons between triangle 1 and triangle 3 are made using similar logic regarding corresponding sides.

Side B from triangle 1 corresponds with M from triangle 3; similarly, side C relates to height h based on facing angles.

Hypotenuses are compared next; A from one triangle faces B in another indicating proportionality.

Finalizing Proportional Relationships

The focus shifts to establishing relationships between triangles two and three by identifying corresponding sides again.

Each proportion is carefully laid out: OC correlates with B; N correlates with H; finally establishing that H correlates with M based on facing angles.

Understanding Triangle Similarity and Metric Relationships

Corresponding Sides and Angles

The speaker emphasizes the importance of corresponding sides in triangles, noting that similarity is determined by matching angles and sides.

Establishing proportional relationships is crucial for understanding triangle similarity; this sets the stage for further calculations.

Deriving Metric Relationships

The speaker aims to explain how metric relationships are derived rather than just presenting them as facts learned from a teacher.

A systematic approach is introduced for calculating proportions between triangle sides, focusing on pairs of comparisons.

Cross-Multiplication Method

The first example involves cross-multiplying to establish a relationship between segments BN and CH, demonstrating the method's application.

Further examples illustrate cross-multiplication with different triangle segments (OA, BC), reinforcing the concept through repetition.

Identifying Repeated Relationships

As relationships are established, the speaker notes repetitions among calculated metrics, emphasizing efficiency in identifying unique relationships.

The process continues with additional comparisons (e.g., BH = CM), highlighting how some results recur throughout calculations.

Pythagorean Theorem Application

The discussion transitions to right triangles, introducing the Pythagorean theorem as a fundamental relationship involving leg lengths and hypotenuse.

Understanding the Pythagorean Theorem

Introduction to Key Concepts

The total length A is defined as the sum of N and m , which will aid in solving exercises later.

The speaker emphasizes the importance of memorizing relationships related to the Pythagorean theorem, indicating that it is a foundational concept already learned.

Deriving the Pythagorean Theorem

Before solving exercises, the speaker introduces an alternative derivation of the Pythagorean theorem using two other metric relations.

By adding two specific relationships, C^2 = AN and B^2 = AM , they aim to demonstrate how these lead to the theorem.

It is noted that since b^2 and c^2 are not similar terms, they cannot be combined directly but can be expressed as b^2 + c^2 .

Factorization and Conclusion

After rearranging terms, it becomes evident that both sides can be factored with a common factor of A .

The relationship shows that combining these metric relations leads back to confirming the Pythagorean theorem: b^2 + c^2 = A^2 .

Application through Exercises

Transitioning into practical application, an exercise is introduced where students must find unknown values in right triangles using established relationships.

For example, calculating side lengths involves applying Pythagoras' theorem: starting with known sides 12 and 16 to find hypotenuse A .

Analyzing Relationships for Unknown Values

To find another unknown value (denoted as h), various relationships are considered based on previously calculated values.

Calculating Unknown Values in Geometry

Understanding the Problem

The speaker discusses why certain values cannot be used in calculations, specifically mentioning the absence of variables M and N.

The calculation begins with A multiplied by height (h), equating to BC. The speaker calculates 12 × 16, arriving at a product of 192.

Performing Division

The division process is explained as the speaker divides 192 by 20, using a short method due to limited space for calculations.

The result of h is determined to be 9.6, but the unit of measurement remains unspecified.

Identifying Unknown Variables

The speaker emphasizes the need to identify which metric relationships can be applied to find unknown values X and h.

A triangle diagram is referenced, indicating vertices A, B, and C while discussing how these relate to finding variable M.

Solving for Variable M

The relationship between sides is explored; however, without known values for N or h, some equations cannot be utilized effectively.

An equation involving b squared equals AM is introduced. With b being known (20), the next step involves solving for M.

Completing Calculations

Squaring b results in an equation: 400 = 25m. To isolate m, divide both sides by 25.

After performing the division (400 ÷ 25), it’s concluded that m equals 16 centimeters.

Finding Additional Unknown N

With m established as 16 cm, the speaker deduces that N must equal 9 to complete a total of 25 when added with m.

Exploring Further Relationships

The discussion shifts back to identifying which relationships can help find additional unknown variables like h and c.

It’s noted that multiple instances of h are present in various equations; however, determining c first may simplify further calculations.

Utilizing Metric Relationships

Understanding the Value of h in Geometry

Calculation of h Squared

The speaker begins by explaining how to find the value of h starting with h^2 . They mention that OM is 16 times N , where N = 9 .

The calculation proceeds as follows: h^2 = 16 times 9 = 144 .

To determine the value of h , the square root of 144 is taken, resulting in h = 12 .

The speaker confirms that this means the value of h in this context is 12 cm.

Application of Pythagorean Theorem

After solving for h , the speaker discusses using the Pythagorean theorem to find other values, specifically mentioning side squared equals hypotenuse squared.

They emphasize that while they could calculate additional measurements, they will focus on what was specifically asked in the exercise.

Encouragement for Practice

The speaker encourages students to practice metric relationships and suggests marking them down for future reference.

They advise tackling exercises one by one to aid memorization and understanding.