Question

Use ruler and compass to answer this question. Construct a triangle ABC where AB = 5.5 cm, BC = 4.5 cm and angle ABC = 135°. Construct the circumcircle to the triangle ABC. Measure and write down the length of AC.

Answer 1

1. Construct triangle ABC

Draw base AB: Use a ruler to draw a line segment AB = 5.5 cm.

Construct angle 135° at B:

1. Extend AB beyond B to form a straight line.

2. Construct a 90° angle at B using arcs bisecting a 180° straight angle.

3. Bisect the adjacent 90° angle the exterior angle to get 45°. Adding this to the. interior 90° or subtracting from 180° results in a 135° angle.

Locate point C: From B, measure 4.5 cm along the 135° line using a compass and mark point C. 

Complete the triangle: Join A and C with a ruler.

2. Construct the circumcircle

Find perpendicular bisectors:

1. Use the compass to draw the perpendicular bisector of AB. 

2. Draw the perpendicular bisector of BC or AC. 

Locate circumcenter (O): The point where these two perpendicular bisectors intersect is the circumcenter O.

Draw the circle: Place the compass point at O and adjust the width to reach vertex A or B or C. Draw the circle passing through all three vertices.

3. Measure AC

Use the ruler to measure the distance between A and C.

Based on the law of cosines b² + a² + c² – 2accos(B), the calculated length is

`AC = sqrt(4.5^2 + 5.5^2 - 2(4.5)(5.5)cos(135^circ)) ≈ 9.25  cm`

The length of AC is approximately 9.25 cm.