Explain why an isosceles triangle is created if the third vertex in on the line y=x

Question

Suppose two vertices of a triangle are (0,x) and (x,0)

Answer 1

Often, it helps to draw a picture.

 

Given the points (0,a) and (a.0):                       [note:  a is a constant; the use of "x" may be confusing]

 

      |

      a

      |

      |

      |

      |

- - - - - - - - - - - - - - - - - - - a -- - -

      |

      |

 

The line y=x will produce the third point (a,a).  The line y=x goes through the origin (0,0) and has a slope of 1 (that is, a 45° angle), so it bisects the line segment formed by the points (0.a) and (a,0).  Any point on this line will be equidistant from (0,a) and (a,0), thus an isosceles triangle is formed.

 

Be careful to exclude the midpoint of the segment because this forms a straight line rather than a triangle.

Answer 2

Because the distance between the two vertices and any point along that line will be the same. However, there is one point along that line where an isosceles triangle won't be formed.