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Questions and Answers
What is the formula used to calculate the area of a triangle when the lengths of all three sides are known?
What is the formula used to calculate the area of a triangle when the lengths of all three sides are known?
What is the semi-perimeter 's' in Heron's Formula?
What is the semi-perimeter 's' in Heron's Formula?
What are the fields in which Heron's Formula is commonly used?
What are the fields in which Heron's Formula is commonly used?
What is an advantage of using Heron's Formula?
What is an advantage of using Heron's Formula?
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What is the first step in using Heron's Formula?
What is the first step in using Heron's Formula?
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What is the formula for the semi-perimeter 's'?
What is the formula for the semi-perimeter 's'?
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What is the unit of the area calculated using Heron's Formula?
What is the unit of the area calculated using Heron's Formula?
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What type of triangles can Heron's Formula be applied to?
What type of triangles can Heron's Formula be applied to?
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What is the final step in using Heron's Formula?
What is the final step in using Heron's Formula?
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What is the general form of a linear equation in 2 variables?
What is the general form of a linear equation in 2 variables?
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What is the graphical representation of a linear equation in 2 variables?
What is the graphical representation of a linear equation in 2 variables?
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What is a solution to a linear equation in 2 variables?
What is a solution to a linear equation in 2 variables?
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What are the methods to find the solutions of linear equations in 2 variables?
What are the methods to find the solutions of linear equations in 2 variables?
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What are the types of linear equations in 2 variables?
What are the types of linear equations in 2 variables?
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What is the real-world application of linear equations in 2 variables?
What is the real-world application of linear equations in 2 variables?
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What is the condition for a and b in the general form of a linear equation in 2 variables?
What is the condition for a and b in the general form of a linear equation in 2 variables?
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What happens to the graph of a linear equation in 2 variables if a = 0?
What happens to the graph of a linear equation in 2 variables if a = 0?
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What is the advantage of using linear equations in 2 variables in real-world problems?
What is the advantage of using linear equations in 2 variables in real-world problems?
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Study Notes
Heron's Formula
Definition
Heron's Formula is a mathematical formula used to calculate the area of a triangle when the lengths of all three sides are known.
Formula
The formula is:
A = √(s(s-a)(s-b)(s-c))
where:
-
Ais the area of the triangle -
a,b, andcare the lengths of the three sides of the triangle -
sis the semi-perimeter, which is half the perimeter of the triangle:s = (a + b + c) / 2
How it Works
- Calculate the semi-perimeter
sby adding the lengths of the three sides and dividing by 2. - Plug the values of
a,b,c, andsinto the formula. - Calculate the area
Aby taking the square root of the product of the three terms.
Example
Given a triangle with sides of length 3, 4, and 5, calculate the area using Heron's Formula.
-
s = (3 + 4 + 5) / 2 = 6 -
A = √(6(6-3)(6-4)(6-5)) = √(6 × 3 × 2 × 1) = √36 = 6
The area of the triangle is 6 square units.
Applications
Heron's Formula is used in various fields, including:
- Geometry and trigonometry
- Physics and engineering
- Computer science and graphics
- Architecture and construction
Advantages
Heron's Formula is useful because it:
- Allows for the calculation of the area of a triangle without knowing the height or angles
- Is applicable to all types of triangles, including scalene, isosceles, and equilateral triangles
- Is a simple and efficient formula to use in calculations
Heron's Formula
Definition
- Heron's Formula is a mathematical formula that calculates the area of a triangle when the lengths of all three sides are known.
Formula
- The formula is:
A = √(s(s-a)(s-b)(s-c)) -
Ais the area of the triangle -
a,b, andcare the lengths of the three sides of the triangle -
sis the semi-perimeter, which is half the perimeter of the triangle:s = (a + b + c) / 2
How it Works
- Calculate the semi-perimeter
sby adding the lengths of the three sides and dividing by 2 - Plug the values of
a,b,c, andsinto the formula - Calculate the area
Aby taking the square root of the product of the three terms
Example
- Given a triangle with sides of length 3, 4, and 5, calculate the area using Heron's Formula
-
s = (3 + 4 + 5) / 2 = 6 -
A = √(6(6-3)(6-4)(6-5)) = √(6 × 3 × 2 × 1) = √36 = 6 - The area of the triangle is 6 square units
Applications
- Geometry and trigonometry
- Physics and engineering
- Computer science and graphics
- Architecture and construction
Advantages
- Allows for the calculation of the area of a triangle without knowing the height or angles
- Is applicable to all types of triangles, including scalene, isosceles, and equilateral triangles
- Is a simple and efficient formula to use in calculations
Linear Equations in 2 Variables
Definition
- Linear equation in 2 variables takes the form
ax + by = c -
a,b, andcare constants, whilexandyare variables -
aandbcannot both be zero
Graphical Representation
- Graph of a linear equation in 2 variables is a straight line
- Line can slope upward or downward depending on the signs of
aandb - Line can be horizontal if
b = 0or vertical ifa = 0
Solution of Linear Equations
- Solution is an ordered pair
(x, y)that satisfies the equation - Linear equation in 2 variables has infinitely many solutions
- Solutions can be found using:
- Graphical method: finding the point of intersection of two or more lines
- Substitution method: substituting the value of one variable into the equation
- Elimination method: adding or subtracting the equations to eliminate one variable
Types of Linear Equations
- Consistent equations: have a unique solution
- Inconsistent equations: have no solution
- Dependent equations: have infinitely many solutions
Applications
- Linear equations in 2 variables are used to model real-world problems, such as:
- Cost and revenue analysis
- Supply and demand analysis
- Motion along a straight line
- Work and time problems
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Description
Calculate the area of a triangle using Heron's Formula. Learn the formula and how to apply it to find the area of a triangle given the lengths of its sides.
