Geometrical Constructions

A becomes equidistant from P and Q

Step 1: In the construction, we create points P and Q using equal arcs from centers A and B. Step 2: By construction, AP = BP (equal radii from A) and AQ = BQ (equal radii from B). Step 3: This means A is equidistant from both P and Q. Step 4: Similarly, B is also equidistant from P and Q, which is why line PQ is the perpendicular bisector of AB.

90° angle

Step 1: To construct 45°, we first construct a 90° angle (right angle). Step 2: Then we bisect this 90° angle using the angle bisector construction. Step 3: Bisecting means dividing into two equal parts: 90° ÷ 2 = 45°. Step 4: This gives us the required 45° angle.

On side AB such that AD = AC

Step 1: Since AB > AC and we need AB - AC = 1.6 cm, we mark D on AB. Step 2: We make AD = AC, which means D divides AB such that the remaining part BD represents the difference. Step 3: Now BD = AB - AD = AB - AC = 1.6 cm. Step 4: We then use perpendicular bisector of CD to locate vertex A correctly.

180°

Step 1: Given angle B = 60° and angle C = 45°. Step 2: Using angle sum property of triangle: A + B + C = 180°. Step 3: Therefore, angle A = 180° - 60° - 45° = 75°. Step 4: The sum is always 180° for any triangle, which validates our construction.