Properties of Triangle - Projection Formula

Duration: 13 min

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  1. About the Course: About the Course
  2. Section 1 | Language (Hindi): भाषा, लिपि, ध्वनि, वर्ण, अक्षर, स्वर, व्यंजन, मात्रा और संयुक्ताक्षर, वर्तनी, वर्ण-विन्यास, अनुस्वार, अनुनासिक, चंद्रबिंदु और ‘र’ के रूप, शब्द, पद और शब्द-भेद, संज्ञा और सर्वनाम, विशेषण और क्रिया, लिंग, वचन, पुरुष और कारक, काल, वाच्य, वाक्य और वाक्य-भेद, संधि, समास, विलोम शब्द, पर्यायवाची शब्द, सामान्य अशुद्धियाँ, अनेक शब्दों के लिए एक शब्द, मुहावरे और लोकोक्तियाँ, अपठित गद्यांश और अपठित पद्यांश, Multi-Topic Content (Cross-Concept Material), वर्णमाला
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  10. Section 3 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Threads & Process Creation, Memory Management, Virtual Memory, Disc Scheduling, File Management, Process Synchronization
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  12. Section 3 | Data Structures: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Minimum Spanning Trees, Shortest Path Algos
  13. Section 3 | Theory Of Computation & Compiler Design: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines, Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization
  14. Section 3 | Emerging Trends (AI, Societal Ethics & MS Office): Artificial Intelligence, Emerging Tech Trends, Society, Law & Ethics, MS Office Applications
  15. Section 3 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Maintenance and Configuration Management, Software Testing
  16. Section 3 | C Programming: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  17. Section 3 | OOPS, C++, JAVA, HTML: OOPS with C++, Java, HTML Basics, CSS
  18. Section 3 | Java: Java Fundamentals, Array, String, IO & References
  19. Section 3 | Python: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis
  20. Section 3 | Digital Electronics: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation
  21. Section 3 | Multimedia: Multimedia
  22. BPSC Previous Year Papers:
  23. BPSC Full Mock Tests:
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This video is a mathematics lecture that derives the projection formula for a triangle. The instructor begins by introducing the concept and drawing a triangle ABC, labeling its vertices and sides. He then drops a perpendicular from vertex A to the opposite side BC, creating a point D. Using the right-angled triangles ABD and ADC, he applies the definition of cosine to find the lengths of the segments BD and DC in terms of the sides and angles of the main triangle. This leads to the derivation of the projection formula: a = b cos C + c cos B. The lecture then presents two example problems. The first problem asks for the value of (cos C + cos A)/c + a + cos B/b, which is solved by substituting the projection formulas. The second problem involves finding the value of b cos²(C/2) + c cos²(B/2) given the perimeter k, which is solved using the half-angle identity for cosine.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video starts with a blank screen, then the instructor begins writing the title "Projection Formula:". He draws a triangle ABC, labeling the vertices and the sides opposite to them as a, b, and c. He then draws a perpendicular from vertex A to the base BC, labeling the foot of the perpendicular as D. The instructor labels the segments BD and DC, and the side BC as a, establishing the relationship a = BD + DC.

  2. 2:00 – 5:00 02:00-05:00

    The instructor analyzes the right-angled triangle ABD. He writes the formula for cosine of angle B: cos B = BD/c, which he rearranges to find BD = c cos B. He then moves to the right-angled triangle ADC, writing cos C = DC/b, which he rearranges to find DC = b cos C. He substitutes these expressions into the earlier equation a = BD + DC, resulting in the projection formula: a = c cos B + b cos C.

  3. 5:00 – 10:00 05:00-10:00

    The instructor writes down the three projection formulas for triangle ABC: a = b cos C + c cos B, b = c cos A + a cos C, and c = a cos B + b cos A. He then presents the first example problem, Eq 1, which asks to find the value of (cos C + cos A)/c + a + cos B/b. He substitutes the projection formulas into the expression, simplifying it to (b cos C + c cos B + c cos A + a cos C + a cos B + b cos A) / (a + b + c). After further simplification, he arrives at the final answer, which is 1.

  4. 10:00 – 13:10 10:00-13:10

    The instructor presents the second example problem, Eq 2. It states that if k is the perimeter of triangle ABC, then what is the value of b cos²(C/2) + c cos²(B/2). He writes the half-angle identity cos²(θ/2) = (1 + cos θ)/2. He applies this to the expression, writing b cos²(C/2) + c cos²(B/2) = b(1 + cos C)/2 + c(1 + cos B)/2. He then expands this to (b + b cos C + c + c cos B)/2. Using the projection formula a = b cos C + c cos B, he substitutes to get (b + c + a)/2. Since k = a + b + c, the expression simplifies to k/2.

The video provides a comprehensive derivation of the projection formula for a triangle, starting from basic trigonometric principles in right-angled triangles. The core of the lesson is the derivation of the formula a = b cos C + c cos B, which is then used as a powerful tool to solve two distinct problems. The first problem is a direct algebraic manipulation of the formula, while the second problem requires the application of a trigonometric identity (the half-angle formula for cosine) in conjunction with the projection formula. The progression demonstrates how a fundamental geometric identity can be applied to solve both algebraic and trigonometric problems in triangle geometry.

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