# Three parallels

The vertices of an equilateral triangle lie on 3 different parallel lines. The middle line is 5 m and 3 m distant from the end lines. Calculate the height of this triangle.

Result

v =  7 m

#### Solution:

$a^2 = 3^2 +x^2 \ \\ a^2 = 5^2 + y^2 \ \\ a^2 = 8^2 + (y-x)^2 \ \\ \ \\ 3^2 +x^2 = 5^2 + y^2 \ \\ 3^2 +x^2 = 8^2 + (y-x)^2 \ \\ \ \\ 3^2 +x^2 = 8^2 + y^2-2xy +x^2 \ \\ 3^2 = 8^2 + y^2-2xy \ \\ \ \\ x = (8^2 - 3^2 + y^2) / (2y) \ \\ \ \\ y = 11 / \sqrt{ 3 } \doteq 6.3509 \ m \ \\ \ \\ x = (8^2 - 3^2 + y^2) / (2y) = (8^2 - 3^2 + 6.3509^2) / (2 \cdot \ 6.3509) \doteq 7.5056 \ m \ \\ \ \\ a = \sqrt{ 3^2 + x^2 } = \sqrt{ 3^2 + 7.5056^2 } \doteq 8.0829 \ m \ \\ \ \\ v^2 = a^2 - (a/2)^2 \ \\ \ \\ v = \sqrt{ a^2 - (a/2)^2 } = \sqrt{ 8.0829^2 - (8.0829/2)^2 } = 7 = 7 \ \text { m }$

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