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2abcc=2abc>2

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12y ago

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What is proof of Heron's Formula?

This is a proof that uses the cosine rule and Pythagoras' theorem. As on any triangle with c being the opposite side of θ and a and b are the other sides: c^2=a^2+b^2-2abcosθ We can rearrange this for θ: θ=arccos[(a^2+b^2-c^2)/(2ab)] On a right-angle triangle cosθ=a/h. We can therefore construct a right-angle triangle with θ being one of the angles, the adjacent side being a^2+b^2-c^2 and the hypotenuse being 2ab. As the formula for the area of a triangle is also absinθ/2, when a and b being two sides and θ the angle between them, the opposite side of θ on the right-angle triangle we have constructed is 4A, with A being the area of the original triangle, as it is 2absinθ. Therefore, according to Pythagoras' theorem: (2ab)^2=(a^2+b^2-c^2)^2+(4A)^2 4a^2*b^2=(a^2+b^2-c^2)^2+16A^2 16A^2=4a^2*b^2-(a^2+b^2-c^2)^2 This is where it will start to get messy: 16A^2=4a^2*b^2-(a^2+b^2-c^2)(a^2+b^2-c^2) =4a^2*b^2-(a^4+a^2*b^2-a^2*c^2+a^2*b^2+b^4-b^2*c^2- a^2*c^2-b^2*c^2+c^4) =4a^2*b^2-(a^4+2a^2*b^2-2a^2*c^2+b^4-2b^2*c^2+c^4) =-a^4+2a^2*b^2+2a^2*c^2-b^4+2b^2*c^2-c^4 (Eq.1) We will now see: (a+b+c)(-a+b+c)(a-b+c)(a+b-c) =(-a^2+ab+ac-ab+b^2+bc-ac+bc+c^2)(a^2+ab-ac-ab-b^2+bc+ac+bc-c^2) =(-a^2+b^2+2bc+c^2)(a^2-b^2+2bc-c^2) =-a^4+a^2*b^2-2a^2*bc+a^2*c^2+a^2*b^2-b^4+2b^3*c-b^2*c^2+2a^2*bc-2b^3*c+(2bc)^2-2bc^3+a^2*c^2-b^2*c^2+2bc^3-c^4 =-a^4+2a^2*b^2+2a^2*c^2-b^4+(2bc)^2-c^4-2b^2*c^2 =-a^4+2a^2*b^2+2a^2*c^2-b^4+2b^2*c^2-c^4 (Eq.2) And now that we know that Eq.1=Eq.2, we can make Eq.1=(a+b+c)(-a+b+c)(a-b+c)(a+b-c) Therefore: 16A^2=(a+b+c)(-a+b+c)(a-b+c)(a+b-c) A^2=(a+b+c)(-a+b+c)(a-b+c)(a+b-c)/16 =[(a+b+c)/2][(-a+b+c)/2][(a-b+c)/2][(a+b-c)/2] And so if we let s=(a+b+c)/2 A^2=s(s-a)(s-b)(s-c)


What did the pythagorean theorm do for math?

If you know two sides of a right angle triangle, you can figure out the third by using the following formula: A*A+B*B=C*C or A**2+B**2=C**2 or C*C-B*B=A*A or C**2-B**2=A**2 or C*C-A*A=B*B or C**2-A**2=B**2 A is the bottom side if the right angle is at the bottom left, B is The only vertical(strait up and down) line if A is correct, and C is the last line.


Show that among all rectangles with area A the square has the minimum perimeter?

Suppose sqrt(A) = B ie the square with sides B has an area of A and its perimeter is 4*B. Now consider a rectangle with sides C and D whose area is A. So C*D = A = B*B so that D = B*B/C Perimeter of the rectangle = 2*(C+D) = 2*C + 2*D = 2*C +2*B*B/C Now consider (C-B)2 which, because it is a square, is always >= 0 ie C*C + B*B - 2*B*C >= 0 ie C*C + B*B >= 2*B*C Multiply both sides by 2/C (which is >0 so the inequality remains the same) 2*C + 2*B*B/C >= 4*B But, as shown above, the left hand side is perimeter of the rectangle, while the right hand side is the perimeter of the square.


Rearrange a equals 2b - c to make b the subject?

a = 2b - c a + c = 2b (a+c)/2 = b b = (a+c)/2


How do you complete a square in a math problem?

Here are the steps: ax^2 + bx + c = 0 Subtract c and divide by a x^2 + (b/a)x = -(c/a) Take the square of (b/a)/2 and add it to both sides (x + ((b/a)/2))^2 = -(c/a) + ((b/a)/2)^2 Take the square root of both sides Subtract ((b/a)/2) and you have your solutions: x = -(c/a) + ((b/a)/2)^2 - ((b/a)/2) x = (c/a) - ((b/a)/2)^2 - ((b/a)/2)


Evaluate a - b - c if a equals -15 b equals 5 and c equals -2?

If a = -15, b = 5 and c = -2 a - b - c = -15 - 5 - (-2) = -20 + 2 = -18


How do you find a leg in a right triangle?

In a right the triangle with legs a, b and hypotenuse c, a^2 = c^2 - b^2 or b^2 = c^2 - a^2.


What is the formula for a Venn Diagram?

A or B or C = A + B + C - A and B - A and C - B and C - 2 (A and B and C) I'm not sure by the way;


A minus b plus c whole square?

If you mean (a-b+c)^2, then... a^2 - ab + ac - ab + b^2 - bc + ac - bc + c^2 = a^2 + b^2 + c^2 - 2ab + 2ac - 2bc.


Asquare plus b square plus c square -ab -bc -ca equals 0 then show that a equals b equals c?

a^2 + b^2 + c^2 - ab - bc - ca = 0=> 2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca = 0 => a^2 - 2ab + b^2 + b^2 - 2bc + c^2 + c^2 - 2ca + a^2 = 0 => (a - b)^2 + (b - c)^2 + (c - a)^2 = 0 Each term on the left hand side is a square and so it is non-negative. Since their sum is zero, each term must be zero. Therefore: a - b = 0 => a = b b - c = 0 => b = c.


What is (a-b plus c)2?

The Pythagoream Thereom is a^2 + b^2 = c^2. Written out it is a squared plus b squared equals c squared.


What is the surface area of a cuboid?

The answer should be: (2*a*b)+(2*b*c)+(2*c*a)