The perpendicular bisector of the of the hypotenuse side forms a right
triangle with the hypotenuse and the longer leg of the of the triangle.
The ratio, of the length of the segment cut from the perpendicular bisector by the hypotenuse and the longer leg to the length of the longer leg, is 1 : 3
Reason:
The given parameters are;
An acute angle of the right triangle = 30°
Let R represent the length of the hypotenuse side, we have;
The length of the longer leg, [tex]L_{ll}[/tex] = R × cos(30°) = [tex]\dfrac{\sqrt{3} }{2} \cdot R[/tex]
The perpendicular bisector forms a right triangle with a leg length = R/2
The length of the hypotenuse formed by the bisector is therefore;[tex]R_{bisector} = \dfrac{\dfrac{R}{2} }{cos(30^{\circ})} = \dfrac{\dfrac{R}{2} }{\dfrac{\sqrt{3} }{2} } = \dfrac{R}{\sqrt{3} }[/tex]
The length, L, of the segment cut by the hypotenuse and the longer leg is therefore;
[tex]L= R_{bisector} \times sin(30^{\circ}) = \dfrac{1}{2} \times \dfrac{R}{\sqrt{3} } = \dfrac{R}{2 \cdot \sqrt{3} }[/tex]
The ratio, of the length of the segment cut from this perpendicular bisector by the hypotenuse and the longer leg to the length of the longer leg, r, is therefore;
[tex]r = \dfrac{L}{L_{ll}} = \dfrac{ \dfrac{R}{2 \cdot \sqrt{3} }}{\dfrac{\sqrt{3} }{2} \cdot R} = \dfrac{2 \cdot R}{2 \cdot \sqrt{3} \times \sqrt{3} \times R } = \dfrac{1}{3}[/tex]
The ratio, of the lengths, r = [tex]\dfrac{1}{3}[/tex], therefore, r = 1:3
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