To construct ΔDEF with the given information, follow these steps:
1. Draw a line segment DE of length 6 cm.
2. At point D, construct an angle of 120 degrees using a protractor. This angle will be angle DEF.
3. At point E, construct an angle of 22.5 degrees. This angle will be angle EDF.
4. Draw the line segment DF to complete the triangle ΔDEF.
To measure the lengths DF and EF, use a ruler:
- Measure DF by placing the ruler at points D and F and reading the length of the segment.
- Measure EF by placing the ruler at points E and F and reading the length of the segment.
Now let's move on to constructing the loci and finding their intersections:
1. Locus l1: To construct the locus of points equidistant from DF and DE, use a compass. Set the compass to the distance between DF and DE. Place the compass at point D and draw an arc that intersects the line segment DE. Repeat the process with the compass centered at point E and draw another arc intersecting the line segment DE. The points where the arcs intersect on line DE will be part of locus l1.
2. Locus l2: To construct the locus of points equidistant from FD and FE, use a compass. Set the compass to the distance between FD and FE. Place the compass at point F and draw an arc that intersects the line segment DE. Repeat the process with the compass centered at point E and draw another arc intersecting the line segment DE. The points where the arcs intersect on line DE will be part of locus l2.
3. Locus l3: To construct the locus of points equidistant from D and F, use a compass. Set the compass to the distance between points D and F. Place the compass at point D and draw an arc. Repeat the process with the compass centered at point F and draw another arc. The points where the arcs intersect will be part of locus l3.
Find the points of intersection of l1, l2, and l3. The point of intersection will be labeled as point P.
Lastly, to draw the incircle, use point P as the center. With the compass set to any radius, draw a circle that intersects the sides of the triangle ΔDEF. Measure PE and PF by placing the ruler on the circle and reading the lengths of the segments.
Note: The exact measurements of DF, EF, PE, and PF can only be determined by performing the construction accurately.