Constructions

The right angle

Step 1: In RHS (Right angle-Hypotenuse-Side) construction, we have the right angle, hypotenuse length, and one other side. Step 2: First draw the given side, then construct a 90° angle at one end of this side. Step 3: From the other end of the given side, draw an arc with radius equal to the hypotenuse. Step 4: This arc intersects the 90° line to complete the right triangle. The right angle must be constructed first because it defines the perpendicular direction for completing the triangle.

The tangent line

Step 1: A tangent to a circle is always perpendicular to the radius at the point of tangency. Step 2: To construct a tangent at point P on circle with center O, first join OP. Step 3: At point P, construct a perpendicular line to radius OP. Step 4: This perpendicular line is the required tangent. This works because the tangent touches the circle at exactly one point and makes a 90° angle with the radius at that point.

Because of the angle bisector property

Step 1: When we draw perpendicular bisectors of sides from the apex to the base, these bisectors create angle bisectors. Step 2: The angle bisector divides the original angle into two equal parts (half-angles). Step 3: By constructing half of each base angle at the ends of the perimeter line, we ensure the perpendicular bisectors meet correctly. Step 4: This geometric property ensures that when we locate the base points using perpendicular bisectors, the final triangle has the required base angles and perimeter.

The included side between the two angles

Step 1: SAS stands for Side-Angle-Side construction method. Step 2: The middle 'S' represents the side that lies between the two given angles. Step 3: This included side is crucial because it connects the two vertices where the given angles are constructed. Step 4: To construct using SAS: draw the included side, then construct the given angles at both ends of this side. The intersection of the angle rays forms the third vertex. The included side is essential for uniquely determining the triangle's shape.