CHAPTER 2: Geometric Constructions

Geometric Constructions Contents Points and Lines Angles Triangles Quadrilaterals Polygons Circles and Arcs Solids Constructions of basic geometrics

Geometric Constructions Points and Lines A point represents a location in space. A line is the shortest distance between two points.

Geometric Constructions Angles An angle is formed by two intersecting lines.

Geometric Constructions Triangles A triangle is a plane figure bounded by three straight sides. The sum of the interior angles is always 180º.

Geometric Constructions Quadrilaterals A quadrilateral is a plane figure bounded by four straight sides. If the opposite sides are parallel, the quadrilateral is also a parallelogram.

Geometric Constructions Polygons A Polygon is any plane figure bounded by straight sides. If the polygon has equal angles and equal sides it can be inscribed in or circumscribed around a circle and is called a regular polygon.

Geometric Constructions Circles and Arcs A circle is a closed curve, all points of which are the same distance from a point called the center.

Geometric Constructions Solids Solids bounded by plane surfaces are called polyhedra. The surfaces are called faces. If the faces are equal regular polygons the solids are called regular polyhedra.

Geometric Constructions Bisecting a line or circular arc Given the line or arc (AB) to be bisected. From (A) and (B) draw equal arcs with radius greater than half of (AB). Join the intersections (D) and (E) with a straight line to locate center (C).

Geometric Constructions Bisecting an angle Given the angle (BAC) to be bisected. Strike the large arc (R) at any convenient radius. Strike equal arcs (r) with a radius slightly larger than half (BC) to intersect at (D). Draw line (AD) which bisects the angle.

Geometric Constructions Transferring an angle Given the angle (BAC) to be transferred to the new position at (A’B’). Use any convenient radius (R) and strike arcs from centers (A) and (A’). Strike equal arcs (r) and draw side (A’C’).

Geometric Constructions Drawing a line parallel to a line at a given distance Given the line (AB) and the distance (CD) Draw two arcs at (E) and (F) with a radius equal to (CD). Draw the line (GH) tangent to the two arcs.

Geometric Constructions Dividing a line into equal parts Given the line to be divided Draw a light construction line at any convenient angle from one end of the given line. With dividers or scale, set off from the intersections of the lines as many equal divisions as needed (in this example, three). Connect the last division point to the other end of the given line using a triangle and T-square. Slide the triangle along the T-square and draw parallel lines through the other division points.

Geometric Constructions Drawing a Square Given the inscribed circle, draw two diameters at right angles to each other. The intersections of these diameters with the circle are the vertexes of an inscribed square. Given the circumscribed circle, use the T-square and 45º triangle and draw the four sides tangent to the circle.

Geometric Constructions Drawing a Hexagon Given the inscribed circle, draw vertical and horizontal center lines and the diagonals (AB) and (CD) at 30º or 60º with the horizontal. With the 30º x 60º triangle and T-square, draw the six sides of the hexagon. Given the circumscribed circle, draw vertical and horizontal center lines . With the 30º x 60º triangle and T-square, draw the six sides of the hexagon tangent to the circle.

Geometric Constructions Drawing an Arc Tangent to a Line and Through a Point Given line (AB), point (P), and radius (R) Draw line (DE) parallel to the given line (AB) at the distance (R) from it. From point (P) draw an arc with a radius (R) intersecting line (DE) at point (C). From point (C) draw the arc tangent to line (AB) and through point (P).

Geometric Constructions Drawing an Arc Tangent to Two Lines at Acute or Obtuse Angles Given two intersecting lines not making a 90º angle and the distance (R) Draw lines parallel to the given lines at a distance (R) from them to intersect at point (C). With (C) as the center and with the given radius (R) draw the required tangent arcs between the given lines.

Geometric Constructions Drawing an Arc Tangent an Arc and a Straight Line Given the straight line (AB) and the arc with radius (G) and the distance (R) Draw a line parallel to the given lines at the distance (R). Draw an arc from center (O) with a radius equal to (G) plus (R) to intersect at (C). With (C) as the center and with the given radius (R) draw the required arc at the given radius (R) and tangent to the given line and arc.

Geometric Constructions Drawing an Arc Tangent to Two Arcs Given the two arcs with centers (A) and (B) and the distance (R) With (A) and (B) as centers draw arcs parallel to the given arcs at the distance (R) from them to locate the intersection (C). With (C) as the center draw the required tangent arc at the given radius (R) to the given arcs.