Question

Using a compass and a straightedge, a student constructed a triangle in which ¯¯¯¯¯¯ X Y is one of the sides.

The compass is opened to a set length, and two intersecting arcs are drawn above ¯¯¯¯¯¯ X Y using X and Y as the centers. The intersection of the two arcs is labeled as point Z. What could be the set length of the compass so that △XYZ is isosceles but not equilateral?

Answer 1

To construct an isosceles but not equilateral triangle △XYZ, a student can open the compass to a set length where the intersecting arcs above ¯¯¯¯¯¯ XY form an unequal-sided triangle.

For △XYZ to be isosceles but not equilateral, the side ¯¯¯¯¯¯ XY must be unequal to ¯¯¯¯¯¯ XZ and ¯¯¯¯¯¯ YZ. This can be achieved by choosing a set length for the compass such that the arc intersections form a triangle with two sides of equal length.

Let's consider a specific case: Open the compass to a length where the arcs intersect above ¯¯¯¯¯¯ XY at point Z, creating an isosceles triangle. However, ensure that the length of ¯¯¯¯¯¯ XZ is not equal to ¯¯¯¯¯¯ YZ. This can be done by choosing a length such that the distance from X to Z is different from the distance from Y to Z, resulting in an isosceles but not equilateral △XYZ. The specific length would depend on the scale and proportions desired for the construction.