You are given two segments, AB with AB = 4 and CD with CD = 7. You want to construct a new segment, EF with EF = 1. If you can use only congruent-segments construction, explain how you would use AB and CD to construct EF.
Draw a line segment of sufficient length
Set your compass to length AB.
Mark off two segments of length AB on the new line. Call the endpoints E and G. EG has length 4+4=8.
Set the compass to the length of CD, or 7.
Place the compass at G and mark a point on EG between E and G. Call that F.
GF = 7, and EF = 1
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To construct a segment EF with EF = 1 using only congruent-segment construction, you can follow these steps:
1. Draw segment AB with a length of 4 units.
2. Draw segment CD with a length of 7 units next to AB.
3. Extend segment AB to the right to create point G.
4. Extend segment CD to the right to create point H. The distance between G and H should be equal to 1 unit.
5. Using compass and ruler, construct a segment GH with a length of 1 unit, drawing a line perpendicular to GH from point G.
6. Label the intersection of the perpendicular line with segment GH as point E.
7. Connect point E with point H to form segment EH.
8. Use compass to mark a point on segment AB at a distance of 1 unit from point A. Label this point as F.
9. Drawing a line through points F and E, extend it until it intersects segment CD.
10. Label the intersection point as point I.
11. Construct segment FI to connect points F and I.
12. Segment FI will be congruent with segment EF and will have a length of 1 unit.
Therefore, by following these steps, you can construct a segment EF with EF = 1 using only congruent-segment construction with segments AB and CD.
To construct a segment EF with EF = 1, using only congruent-segments construction, we can follow the following steps:
1. Start by drawing line segments AB and CD with their given lengths of AB = 4 and CD = 7, respectively.
2. Using a compass, place the needle at point A and open the compasses to a length of 1 unit.
3. Without changing the compasses' width, place the needle at point B and draw an arc that intersects line segment AB. Let's call this point X.
4. Now, using the same compass width, place the needle at point C and draw an arc such that it intersects line segment CD. Let's call this point Y.
5. Draw a line segment through points X and Y. This line intersects AB at point E and CD at point F.
6. Finally, measure the length of EF using a ruler to ensure it is equal to 1 unit.
The line segment EF constructed using congruent-segments construction is now complete, with EF = 1 unit.