Answer :
### 6a) Using a ruler and a pair of compasses only, construct:
#### (i) Triangle XYZ with [tex]\( |XY| = 9 \)[/tex] cm, [tex]\( |YZ| = 12 \)[/tex] cm, and [tex]\( |XZ| = 8 \)[/tex] cm.
Steps:
1. Draw Line Segment XY:
- Draw a straight line segment XY that is 9 cm long using a ruler.
2. Draw Arc with Centre Y and Radius 12 cm:
- Place the compass point on Y (one end of the line segment XY) and draw an arc with a radius of 12 cm.
3. Draw Arc with Centre X and Radius 8 cm:
- Without changing the compass width, place the compass point on X (the other end of the line segment XY) and draw an arc intersecting the first arc.
4. Label the Intersection Point of the Arcs as Z:
- The intersection point of the two arcs is point Z. Connect Z to X and Z to Y to form triangle XYZ.
#### (ii) Construct the Perpendicular Bisector of Line XY:
Steps:
1. Draw Arcs Above and Below XY Using Compasses:
- Place the compass point on X and set it wider than half the length of XY. Draw an arc above and below the line segment XY.
- Without changing the compass width, place the compass point on Y and draw another arc above and below the line segment XY. The arcs should intersect the previous arcs.
2. Draw the Perpendicular Bisector:
- Using a ruler, draw a straight line through the intersection points of the arcs. This line is the perpendicular bisector of XY.
#### (iii) Construct the Perpendicular Bisector of Line XZ:
Steps:
1. Draw Arcs Above and Below XZ Using Compasses:
- Place the compass point on X and set it wider than half the length of XZ. Draw an arc above and below the line segment XZ.
- Without changing the compass width, place the compass point on Z and draw another arc above and below the line segment XZ. The arcs should intersect the previous arcs.
2. Draw the Perpendicular Bisector:
- Using a ruler, draw a straight line through the intersection points of the arcs. This line is the perpendicular bisector of XZ.
### 6b)(i) Label the Point of Intersection of the Two Bisectors as T:
Steps:
1. Identify Intersection Point of Perpendicular Bisectors:
- Find the point where the perpendicular bisectors of lines XY and XZ intersect. Label this point as T.
### 6b)(ii) With Point T as Centre, Draw a Circle of Radius 6 cm:
Steps:
1. Set Compass to 6 cm:
- Adjust the compass width to 6 cm.
2. Draw Circle with Centre T:
- Place the compass point on T and draw a circle with a radius of 6 cm.
### 6c) Measure:
#### (i) Distance [tex]\( |TX| \)[/tex]
Steps:
1. Measure Distance from T to X:
- Using a ruler, measure the distance from point T to point X and record the measurement.
#### (ii) Angle XYZ
Steps:
1. Use a Protractor to Measure Angle XYZ:
- Place the midpoint of the protractor at point Y.
- Align one side of the protractor along line YX.
- Read the scale on the protractor where line YZ intersects to determine the measure of angle XYZ and record the measurement.
These steps should help you construct the triangle and other required geometric elements accurately.
#### (i) Triangle XYZ with [tex]\( |XY| = 9 \)[/tex] cm, [tex]\( |YZ| = 12 \)[/tex] cm, and [tex]\( |XZ| = 8 \)[/tex] cm.
Steps:
1. Draw Line Segment XY:
- Draw a straight line segment XY that is 9 cm long using a ruler.
2. Draw Arc with Centre Y and Radius 12 cm:
- Place the compass point on Y (one end of the line segment XY) and draw an arc with a radius of 12 cm.
3. Draw Arc with Centre X and Radius 8 cm:
- Without changing the compass width, place the compass point on X (the other end of the line segment XY) and draw an arc intersecting the first arc.
4. Label the Intersection Point of the Arcs as Z:
- The intersection point of the two arcs is point Z. Connect Z to X and Z to Y to form triangle XYZ.
#### (ii) Construct the Perpendicular Bisector of Line XY:
Steps:
1. Draw Arcs Above and Below XY Using Compasses:
- Place the compass point on X and set it wider than half the length of XY. Draw an arc above and below the line segment XY.
- Without changing the compass width, place the compass point on Y and draw another arc above and below the line segment XY. The arcs should intersect the previous arcs.
2. Draw the Perpendicular Bisector:
- Using a ruler, draw a straight line through the intersection points of the arcs. This line is the perpendicular bisector of XY.
#### (iii) Construct the Perpendicular Bisector of Line XZ:
Steps:
1. Draw Arcs Above and Below XZ Using Compasses:
- Place the compass point on X and set it wider than half the length of XZ. Draw an arc above and below the line segment XZ.
- Without changing the compass width, place the compass point on Z and draw another arc above and below the line segment XZ. The arcs should intersect the previous arcs.
2. Draw the Perpendicular Bisector:
- Using a ruler, draw a straight line through the intersection points of the arcs. This line is the perpendicular bisector of XZ.
### 6b)(i) Label the Point of Intersection of the Two Bisectors as T:
Steps:
1. Identify Intersection Point of Perpendicular Bisectors:
- Find the point where the perpendicular bisectors of lines XY and XZ intersect. Label this point as T.
### 6b)(ii) With Point T as Centre, Draw a Circle of Radius 6 cm:
Steps:
1. Set Compass to 6 cm:
- Adjust the compass width to 6 cm.
2. Draw Circle with Centre T:
- Place the compass point on T and draw a circle with a radius of 6 cm.
### 6c) Measure:
#### (i) Distance [tex]\( |TX| \)[/tex]
Steps:
1. Measure Distance from T to X:
- Using a ruler, measure the distance from point T to point X and record the measurement.
#### (ii) Angle XYZ
Steps:
1. Use a Protractor to Measure Angle XYZ:
- Place the midpoint of the protractor at point Y.
- Align one side of the protractor along line YX.
- Read the scale on the protractor where line YZ intersects to determine the measure of angle XYZ and record the measurement.
These steps should help you construct the triangle and other required geometric elements accurately.
