Answer :
To construct a triangle XYZ and complete the given set of tasks, follow these steps:
### Part 2a: Construct Triangle XYZ
1. Draw the Base Line XY:
- Using a ruler, draw a straight line segment XY of length 8.7 cm.
- Label the endpoints of the line as X and Y.
2. Construct ∠XZY = 45°:
- Place the needle of the compasses on point Y and draw a gentle arc across the intended location of point Z.
- Place the compasses on point Y again and draw another arc intersecting the first arc, forming an intersection point.
- Using the ruler, draw a line from point Y through this intersection mark. This line represents one side of the 45° angle.
3. Draw Lines for |YZ| = |XZ|:
- Set the compass width to any comfortable length (less than 8.7 cm), and draw an arc from point Y.
- Without changing the compass width, draw an arc from point X.
- The intersection of these two arcs indicates the position of point Z, ensuring |YZ| = |XZ|.
4. Label the Point of Intersection:
- Where the two arcs intersect, label this point as Z.
- Confirm that |YZ| = |XZ|, as these arcs are congruent segments.
### Part 2b: Locate M as the Circumcenter
1. Perpendicular Bisectors:
- Draw the perpendicular bisectors of at least two sides of ΔXYZ. For example, bisect line segment XY:
- Using compasses, mark points that are equal distances from both X and Y, forming intersections above and below the line.
- Draw a line through these perpendicular bisector points.
- Repeat for another side, either XZ or YZ.
- The intersection of these perpendicular bisectors locates the center point M, ensuring it is equidistant from X, Y, and Z.
### Part 2c: Draw the Circumcircle
1. Draw the Circle:
- Place the needle of the compasses on point M and adjust the compass to extend to one of the vertices (X, Y, or Z).
- Draw a circle. This circle should pass exactly through all three vertices (X, Y, Z), touching each side of ΔXYZ.
### Part 2d: Measure and Find the Perimeter
1. Measure Radius:
- Measure the radius from point M to point X (or Y or Z) using a ruler.
2. Perimeter:
- Using the known lengths of the triangle segments, compute the perimeter:
- Since |XZ| = |YZ| = |XY|, the perimeter of ΔXYZ is simply three times the length of one side.
- Example Calculation:
[tex]\[ \text{Perimeter of ΔXYZ} = |XY| + |YZ| + |XZ| \\ \text{Perimeter of ΔXYZ} = 8.7 \text{ cm} + 8.7 \text{ cm} + 8.7 \text{ cm} \\ \text{Perimeter of ΔXYZ} = 3 \times 8.7 \text{ cm} = 26.1 \text{ cm} \][/tex]
- The measured radius and the calculated perimeter conclude part 2d.
By closely following these steps, one can accurately construct the requested triangle, find the circumcenter, draw the circumcircle, and compute further measurements.
### Part 2a: Construct Triangle XYZ
1. Draw the Base Line XY:
- Using a ruler, draw a straight line segment XY of length 8.7 cm.
- Label the endpoints of the line as X and Y.
2. Construct ∠XZY = 45°:
- Place the needle of the compasses on point Y and draw a gentle arc across the intended location of point Z.
- Place the compasses on point Y again and draw another arc intersecting the first arc, forming an intersection point.
- Using the ruler, draw a line from point Y through this intersection mark. This line represents one side of the 45° angle.
3. Draw Lines for |YZ| = |XZ|:
- Set the compass width to any comfortable length (less than 8.7 cm), and draw an arc from point Y.
- Without changing the compass width, draw an arc from point X.
- The intersection of these two arcs indicates the position of point Z, ensuring |YZ| = |XZ|.
4. Label the Point of Intersection:
- Where the two arcs intersect, label this point as Z.
- Confirm that |YZ| = |XZ|, as these arcs are congruent segments.
### Part 2b: Locate M as the Circumcenter
1. Perpendicular Bisectors:
- Draw the perpendicular bisectors of at least two sides of ΔXYZ. For example, bisect line segment XY:
- Using compasses, mark points that are equal distances from both X and Y, forming intersections above and below the line.
- Draw a line through these perpendicular bisector points.
- Repeat for another side, either XZ or YZ.
- The intersection of these perpendicular bisectors locates the center point M, ensuring it is equidistant from X, Y, and Z.
### Part 2c: Draw the Circumcircle
1. Draw the Circle:
- Place the needle of the compasses on point M and adjust the compass to extend to one of the vertices (X, Y, or Z).
- Draw a circle. This circle should pass exactly through all three vertices (X, Y, Z), touching each side of ΔXYZ.
### Part 2d: Measure and Find the Perimeter
1. Measure Radius:
- Measure the radius from point M to point X (or Y or Z) using a ruler.
2. Perimeter:
- Using the known lengths of the triangle segments, compute the perimeter:
- Since |XZ| = |YZ| = |XY|, the perimeter of ΔXYZ is simply three times the length of one side.
- Example Calculation:
[tex]\[ \text{Perimeter of ΔXYZ} = |XY| + |YZ| + |XZ| \\ \text{Perimeter of ΔXYZ} = 8.7 \text{ cm} + 8.7 \text{ cm} + 8.7 \text{ cm} \\ \text{Perimeter of ΔXYZ} = 3 \times 8.7 \text{ cm} = 26.1 \text{ cm} \][/tex]
- The measured radius and the calculated perimeter conclude part 2d.
By closely following these steps, one can accurately construct the requested triangle, find the circumcenter, draw the circumcircle, and compute further measurements.
