Help !!!

An equilateral triangle ABC and there is a point out of the triangle O AO = 3 , BO = 8 , CO = 5 . What's the length of AB ?!!

#Geometry

Easy Math Editor

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Comments

There is a formula that can be derived for the side length of the equilateral triangle given distances from a point inside to the vertices. The formula is $\sqrt{(4 \sqrt{3} A + a^2 + b^2 + c^2)/2}$ where A is area of a triangle formed by those distances. In our case the point is outside of equilateral triangle, and the distances form degenerate triangle of area 0. $AB=\sqrt{(3^2 + 5^2 + 8^2)/2} = \boxed{7}$ I think formula for the side length when the point is outside would be: $x = \sqrt{(a^2 + b^2 + c^2-4 \sqrt{3} A)/2}$ Degenerate triangle in our case is a segment of length 8. When we construct equilateral triangles on one end of the segment, one with side length 3 the other with side length 5,connect the vertices to the other end of the segment, we will get the side length AB. $\angle AOB = \angle BOC = 60$ . Using cosine law would get the result. $x^2 = 8^2 + 5^2 - 2 \times 5 \times 8 \times cos(60) = 49$

Maria Kozlowska - 6 years ago

sorry but i didn't understand what's the A ?!!

Mohab Mahdy - 6 years ago

A is this formula is an area of a triangle formed by AO, BO and CO - the formula for triangle area when just side lengths are given is called Heron's formula. In this case the triangle would be just a segment.

Maria Kozlowska - 6 years ago

@Maria Kozlowska – which area ??

Mohab Mahdy - 6 years ago

@Mohab Mahdy – Can you explain it with a design

Mohab Mahdy - 6 years ago

@Mohab Mahdy – I can not post an image for some reason. One way to do it would be to construct equilateral triangles on both sides of the 8 segment - BO. Then you can prove that when points A and C lie on the sides of these new triangles ABC is equilateral. The reason is that $\triangle ABO and \triangle BCO2$ are congruent and $\angle ABC = 60$ .

Maria Kozlowska - 6 years ago

@Mohab Mahdy – A triangle formed with sides 3, 5, and 8 is a straight line that has 0 area.

Niranjan Khanderia - 5 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$