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Chapter 3 CNC Math 53 54 CNC Machining N60G01X3.25F.002 N70G04X0.5 N80X3.35F.05 N90G00X5.0Z0T0101 U Bisect

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Chapter 3 CNC Math 53 54 CNC Machining N60G01X3.25F.002 N70G04X0.5 N80X3.35F.05 N90G00X5.0Z0T0101 U Bisect This sample chapter is for review purposes only. Copyright © The Goodheart-Willcox Co., Inc. All rights reserved. N10G20G99G40 N20G96S800M3 N30G50S4000 N40T0100M8 N50G00X3.35Z1.25T0101 Chapter 3 CNC Math 53 54 CNC Machining N60G01X3.25F.002 N70G04X0.5 N80X3.35F.05 N90G00X5.0Z0T0101 u Bisect. To divide into two equal parts. 01111 N10G20G99G40 u Congruent. Having the same size and shape. N20G96S800M3 N30G50S4000 u Diagonal. Running from one corner of a four-sided figure to the N40T0100M8 N50G00X3.35Z1.25T0101 opposite corner. N60G01X3.25F.002 N70G04X0.5 u Parallel. Lying in the same direction but always the same distance apart. N80X3.35F.05 Chapter 3 u Perpendicular. At a right angle to a line or surface. u Segment. That part of a straight line included between two points. CNC Math u Tangent. A line contacting a circle at one point. u Transversal. A line that intersects two or more lines. Angles An angle () is the figure formed by the meeting of two lines at the Objectives same point or origin called the vertex. See Figure 3-1. Angles are measured Information in this chapter will enable you to: in degrees (n), minutes (`), and seconds (p). A degree is equal to 1/360 of a circle, a minute is equal to 1/60 of 1n, and a second is equal to 1/60 of 1`. u Identify various geometric shapes. There are many types of angles, Figure 3-2. An acute angle is greater u Apply various geometric principles to solve problems. than 0n and less than 90n. An obtuse angle is greater than 90n and less u Solve right triangle unknowns. than 180n. A right angle is exactly 90n. A straight angle is exactly 180n, or a u Apply trigonometric principles to determine coordinate values. straight line. A reflex angle is greater than 180n and less than 360n. An angle can also be described by its relationship to another angle. Technical Terms See Figure 3-3. Adjacent angles are two angles that use a common side. acute angle function right triangle Complementary angles are two angles that equal 90n. Supplementary adjacent angles hypotenuse scalene triangle angles are two angles that equal 180n, or a straight line. angle isosceles triangle secant arc obtuse angle segment Polygons bisect parallel sine chord parallelogram square Polygons are figures with many sides that are formed by line segments. circle perpendicular straight angle Polygons are named according to the number of sides and angles they have. circumference polygons supplementary angles For example, a decagon is a polygon with ten sides; deca comes from the complementary angles proposition tangent Latin word for ten. congruent Pythagorean theorem transversal cosecant quadrilateral triangle Triangles cosine radius trigonometric A triangle is a three-sided polygon. There are a number of types of cotangent rectangle functions triangles, Figure 3-4. A right triangle has a 90n (right) angle. An isosceles diagonal reflex angle trigonometry triangle has two equal sides and two equal angles. An equilateral triangle diameter right angle vertex has three equal sides; all angles are equal (60n). A scalene triangle has equilateral triangle three unequal sides and unequal angles. Geometric Terms E Figure 3-1. Angle defined as EFG. The following is a list of geometric terms and their definitions. These Vertex terms will be used throughout this chapter and the remainder of this 30° textbook. Study them before continuing. F G 53 Chapter 3 CNC Math 55 56 CNC Machining J D S KK K T E Acute Obtuse Right Right Isosceles P H R O M Straight Reflex HH L Figure 3-2. Various types of angles. N Equilateral Scalene T Figure 3-4. Examples of various types of triangles. The sum of the angles in a triangle always equals 35° 180°. V 55° Quadrilaterals A quadrilateral is a polygon with four sides, Figure 3-5. A line drawn from one angle (intersecting corner) of a quadrilateral to the opposite angle U W is called a diagonal. Adjacent Complementary Square A square has four equal sides and four right (90n) angles. The opposite sides of a square are parallel to each other. The diagonals of a square bisect the four angles and each other. The diagonals are equal and perpendicular 100° to each other. The diagonals form congruent angles (equal in size and shape). 80° Rectangle A rectangle is a quadrilateral with equal opposite sides and four right Supplementary angles. The opposite sides are parallel to each other. The diagonals are equal, bisect each other, and create two pairs of congruent triangles. Figure 3-3. Describing an angle in relationship to another angle. Chapter 3 CNC Math 57 58 CNC Machining M S NT DiameterRadius Circumference Square Rectangle L J K K Q R ChordArc Tangent Figure 3-7. Illustrations of various terms relating to a circle. Parallelogram The circumference is the distance around a circle. A chord is a segment that joins any two points on the circumference of Figure 3-5. The three types of quadrilaterals. Quadrilaterals have four interior angles that total 360°. a circle. An arc is a curved portion of a circle. A tangent to a circle is a line that intersects a circle at a single point. Parallelogram For example, as shown in Figure 3-7, Line L is tangent to the circle and A parallelogram is a quadrilateral with equal opposite sides and equal intersects the circle at Point K. opposite angles. The diagonals bisect each other and create two pairs of congruent triangles. Propositions Circles A proposition is a statement to be proved, explained, or discussed. A circle is a set of points, located on a plane, that are equidistant from a Following are a number of geometric propositions. common central point (center point). See Figure 3-6. There are a number of u Opposite angles are equal. When two lines intersect, they form equal terms that are used to describe various aspects of a circle, Figure 3-7. angles. Thus, in Figure 3-8, Angle 1 equals Angle 3, and Angle 2 The diameter is the segment that connects two points on a circle equals Angle 4. and intersects through the center of the circle. The size of a circle is its u Two angles are equal if they have parallel corresponding sides. diameter. Thus, in Figure 3-9, Angle 1 equals Angle 2. The radius is a segment that joins the circle center to a point on the circle circumference. Radius has half the value of diameter. u A line perpendicular to one of two parallel lines is perpendicular to the other line. Thus, in Figure 3-10, Lines R and S are perpendicular to Line T. Figure 3-6. All points defining a circle are equidistant from the center point. H Figure 3-8. Two intersecting lines form four angles with the opposite angles being 1 equal. 24 3 Chapter 3 CNC Math 59 60 CNC Machining u Alternate interior angles are equal. If two parallel lines are u Two angles having their corresponding sides perpendicular are intersected by a third line (transversal), then alternate interior angles equal. Thus, in Figure 3-14, Angle 1 equals Angle 2. are equal to each other. Thus, in Figure 3-11, Angle 3 equals Angle 6, u A line taken from a point of tangency to the center of a circle is and Angle 4 equals Angle 5. perpendicular to the tangent. Thus, in Figure 3-15, Line TW is u Alternate exterior angles are equal. When two parallel lines are perpendicular to Line UV. intersected by a third line (transversal), then alternate exterior angles u Two tangent lines drawn to a circle from the same exterior point are equal to each other. Thus, in Figure 3-11, Angle 1 equals Angle 8, cause the corresponding segments to be equal in length. Thus, in and Angle 2 equals Angle 7. Figure 3-16, Segment ML equals Segment MN. u Corresponding angles are equal. When two parallel lines are intersected by a third line (transversal), then all corresponding angles are equal. Thus, in Figure 3-11, Angle 1 equals Angle 5, Angle 3 equals Angle 7, Angle 2 equals Angle 6, and Angle 4 equals Angle 8. u The sum of the interior angles of a triangle is 180n. Thus, in Figure 3-12, Angle 1 plus Angle 2 plus Angle 3 equals 180n. u The exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles. Thus, in Figure 3-13, Angle 4 equals Angle 1 plus Angle 2. T 1 2 R 4 1 32 S Figure 3-13. Angle 4, an exterior angle, Figure 3-14. Angle 1 is equal to Angle 2 because their 12 equals the sum of Angles 1 and 2, which are corresponding sides are perpendicular to each other. nonadjacent interior angles. Figure 3-9. Two angles with corresponding parallel sides are Figure 3-10. A transversal line equal. perpendicular to one parallel line is perpendicular to the other parallel M line. Lines R and S are parallel. T U V 1 2 R LN 34 2 56 W W S 78 13 Figure 3-11. Two parallel lines intersected by a third line form Figure 3-12. Angles 1, 2, and 3 alternate angles that are equal to each other. Interior Angles 4 form a triangle totaling 180°. Figure 3-15. Line TW, developed from the Figure 3-16. Lines MN and ML are tangent to the and 5 are equal along with 3 and 6. Exterior Angles 1 and 8 are tangency point to the circle’s center, is circle and share an endpoint and are, therefore, equal equal along with 2 and 7.
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