Geometry problem

Question

An equilateral triangle has centre O and side length 1. A straight line through O intersects the triangle at two distinct points P and Q. Find the minimum possible length of PQ.

Answer 1

draw a diagram

assuming the center O is equidistant perpendicularly from the three sides...

P is the top vertex and Q is the midpoint of the bottom horizontal side

therefore PQ is perpendicular to the bottom side forming 2 right triangles

the line segment PQ also bisects angle P into two 30° angles

you now have two 30-60-90 right triangles

choose one of them

the hypotenuse is the side whose length is 1

the side opposite the 30° angle is half of the hypotenuse, or 1/2

you want the measure of the line segment PQ which is the other leg of the triangle

use the Pythagorean Theorem

1^2=(1/2)^2+PQ^2

1=1/4 + PQ^2

1- 1/4=PQ^2

PQ^2=3/4

take the square root of both sides

PQ=√(3/4)

PQ=√3/√4

PQ=√3/2

PQ=1.7320508/2

PQ=0.8660254 (round as needed)