Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Tangents and Tangency
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Tangents and Tangency
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Subject: Technical Drawings
Class: Senior Secondary 1
Term: 3rd Term
Week: 11
Theme: Geometrical Constructions
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Explain the principles and applications of tangency.
Construct tangents to:a. a point on the circumference of a circle.b. two equal and unequal circles.
Construct tangents of arcs:a. arcs to uching externallyb. arcs to uching in ternally Apply the principles of tangency to constructing objects with arcs, lines and circles.
Principle Applied: Collinearity of centers and point of tangency. The distance between centers will be the difference of their radii.
Steps:
1. Draw the two given arcs with centers O1 (radius R1) and O2 (radius R2).
2. From center O1, open a compass to radius (Rr - R1) and draw an arc. (
Note: Rr must be greater than R1 and R2).
3. From center O2, open a compass to radius (Rr - R2) and draw another arc.
4. The intersection of these two arcs (O3) is the center of the required tangent arc.
5. With O3 as center and Rr as radius, draw the tangent arc.
6. The points of tangency lie on the lines connecting O3 to O1 and O3 to O
2. Reasoning: The center of the tangent arc (O3) must be (Rr - R1) distance from O1 and (Rr - R2) distance from O2 for internal tangency, ensuring the larger tangent arc encloses the smaller arcs. 2.
7. Constructing an Arc Tangent to a Line and a Circle (Externally)
Objective: To draw an arc of a given radius (Rr) that touches a straight line and a given circle on their outer surfaces.
Steps:
1. Draw the given straight line and the given circle with center O and radius R.
2. Draw a line parallel to the given straight line at a distance equal to Rr (the radius of the tangent arc).
3. From the center O of the given circle, open a compass to radius (R + Rr) and draw an arc to intersect the parallel line.
4. The intersection point (Oc) is the center of the required tangent arc.
5. From Oc, drop a perpendicular to the given straight line to find the point of tangency (T1).
6. Draw a line from Oc through O to find the point of tangency (T2) on the circle.
7. With Oc as center and Rr as radius, draw the tangent arc between T1 and T
2. Reasoning: The parallel line ensures tangency to the straight line, and the (R + Rr) arc ensures external tangency to the circle. 2.
8. Constructing an Arc Tangent to Two Straight Lines (Fillet or Round)
Objective: To draw an arc of a given radius (Rr) that touches two non-parallel straight lines (e.g., forming an angle).
Steps:
1. Draw the two given straight lines intersecting at point V.
2. Bisect the angle formed by the two lines. This bisector line will contain the center of the tangent arc.
3. Draw a line parallel to one of the given lines at a perpendicular distance equal to Rr (the radius of the tangent arc).
4. The intersection of this parallel line and the angle bisector (Oc) is the center of the required tangent arc.
5. From Oc, drop perpendiculars to both given straight lines to find the points of tangency (T1 and T2).
6. With Oc as center and Rr as radius, draw the tangent arc between T1 and T2. * Reasoning: The angle bisector ensures that the center of the arc is equidistant from both lines, and the parallel line helps locate this center at the specified radius distance. --- Definition of Key Terms: Tangent: A straight line that touches a curve (typically a circle or an arc) at exactly one point, without crossing it.
Point of Tangency: The single point where a tangent line touches a curve.
Tangency: The condition or state of being tangent. It describes the smooth transition between a line and a curve, or between two curves.
Fundamental Principles of Tangency:
1. Radius Perpendicular to Tangent: A radius drawn from the center of a circle to its point of tangency on the circumference is always perpendicular (at 90 degrees) to the tangent line at that point.
2. Collinearity of Centers and Tangency Point: When two circles or an arc and a circle touch each other (tangentially), their centers and the point of tangency always lie on a single straight line.
3. Parallel Tangents: When two equal circles share a common external tangent, the line connecting the centers of the circles is parallel to the common tangent, and the radii drawn to the points of tangency are perpendicular to the common tangent. Construction Methods (Step-by-step reasoning): 2.
1. Constructing a Tangent to a Point on the Circumference of a Circle Objective: To draw a straight line that touches a circle at a specific given point (P) on its perimeter.
Principle Applied: The radius to the point of tangency is perpendicular to the tangent.
Steps:
1. Draw the given circle with center O and the given point P on its circumference.
2. Draw a line (radius) from the center O to the point P.
3. At point P, construct a line perpendicular to the radius O
P. This can be done by: Opening a compass to a convenient radius, placing the needle at P, and marking two arcs (A and B) on the line OP (or extending OP beyond P if necessary). With the compass needle at A, open it to a radius greater than AP and draw an arc above P. With the same compass radius, place the needle at B and draw another arc to intersect the first arc at point
C. Draw a straight line through C and
P. This line is the required tangent.
Reasoning: This method directly applies the principle that the tangent is perpendicular to the radius at the point of tangency. 2.
2. Constructing a Common External Tangent to Two Equal Circles Objective: To draw a line that touches the outer surfaces of two circles of the same radius.
Principle Applied: Radii to points of tangency are perpendicular to the tangent, and the line connecting centers is parallel to the tangent.
Steps:
1. Draw the two equal circles with centers O1 and O2 and radius R.
2. Draw a straight line connecting the centers O1 and O2.
3. Draw a radius from O1 perpendicular to the line O1O2, marking point P1 on the circumference.
4. Draw a radius from O2 perpendicular to the line O1O2, marking point P2 on the circumference. Ensure P1 and P2 are on the same "side" relative to the line O1O2 (e.g., both above or both below).
5. Draw a straight line connecting P1 and P
2. This line is the common external tangent.
Reasoning: Since the radii O1P1 and O2P2 are parallel and equal, and both are perpendicular to the line O1O2, the line P1P2 will be parallel to O1O2 and tangent to both circles. 2.
3. Constructing a Common External Tangent to Two Unequal Circles Objective: To draw a line that touches the outer surfaces of two circles with different radii.
Principle Applied: Similar triangles and auxiliary circles.
Steps:
1. Draw the two circles with centers O1 (radius R1, larger) and O2 (radius R2, smaller).
2. Draw a straight line connecting the centers O1 and O2.
3. From the center of the larger circle (O1), draw a concentric auxiliary circle with radius (R1 - R2).
4. Bisect the line O1O2 to find its midpoint M.
5. With M as center and MO1 (or MO2) as radius, draw a semicircle that intersects the auxiliary circle at point X.
6. Draw a line from O1 through Similar triangles and auxiliary circles.
Steps:
1. Draw the two circles with centers O1 (radius R1, larger) and O2 (radius R2, smaller).
2. Draw a straight line connecting the centers O1 and O2.
3. From the center of the larger circle (O1), draw a concentric auxiliary circle with radius (R1 - R2).
4. Bisect the line O1O2 to find its midpoint M.
5. With M as center and MO1 (or MO2) as radius, draw a semicircle that intersects the auxiliary circle at point X.
6. Draw a line from O1 through X. Extend this line to touch the circumference of the larger circle at point T
1. This line O1T1 is a radius of the larger circle.
7. From O2, draw a line parallel to O1X towards the circumference of the smaller circle, marking point T2 where it touches.
8. Draw a straight line connecting T1 and T
2. This is the common external tangent.
Reasoning: The construction essentially reduces the problem to drawing a tangent from a point (O2) to a new circle (the auxiliary circle) and then translating this tangent to the original circles. 2.
4. Constructing a Common Internal Tangent to Two Unequal Circles Objective: To draw a line that passes between two circles of different radii, touching their inner surfaces.
Principle Applied: Similar triangles and auxiliary circles.
Steps:
1. Draw the two circles with centers O1 (radius R1, larger) and O2 (radius R2, smaller).
2. Draw a straight line connecting the centers O1 and O2.
3. From the center of the larger circle (O1), draw a concentric auxiliary circle with radius (R1 + R2).
4. Bisect the line O1O2 to find its midpoint M.
5. With M as center and MO1 (or MO2) as radius, draw a semicircle that intersects the auxiliary circle at point X.
6. Draw a line from O1 through X. Extend this line to touch the circumference of the larger circle at point T1.
7. From O2, draw a line parallel to O1X, but in the opposite direction from O1X, towards the circumference of the smaller circle, marking point T2 where it touches.
8. Draw a straight line connecting T1 and T
2. This is the common internal tangent.
Reasoning: Similar to external tangency, but the auxiliary circle uses the sum of radii, allowing the tangent to cross between the circles. 2.
5. Constructing Tangent Arcs (Connecting Arcs Externally)
Objective: To draw an arc of a given radius (Rr) that touches two existing arcs (R1, R2) on their outer sides.
Principle Applied: Collinearity of centers and point of tangency. The distance between centers will be the sum of their radii.
Steps:
1. Draw the two given arcs with centers O1 (radius R1) and O2 (radius R2).
2. From center O1, open a compass to radius (R1 + Rr) and draw an arc.
3. From center O2, open a compass to radius (R2 + Rr) and draw another arc.
4. The intersection of these two arcs (O3) is the center of the required tangent arc.
5. With O3 as center and Rr as radius, draw the tangent arc.
6. The points of tangency lie on the lines connecting O3 to O1 and O3 to O
2. Reasoning: The center of the tangent arc (O3) must be (R1 + Rr) distance from O1 and (R2 + Rr) distance from O2 for external tangency. 2.
6. Constructing Tangent Arcs (Connecting Arcs Internally)
Objective: To draw an arc of a given radius (Rr) that touches two existing arcs (R1, R2) on their inner sides (i.e., the tangent arc encloses the existing arcs).
Principle Applied: Collinearity of centers and point of tangency. The distance between centers will be the difference of their radii. * Steps:
1. Draw the two given arcs with centers O1 (radius R1) and O2 (radius R2).
2. From center O1, open a compass to radius (Rr - R1) and draw an arc. (
Note: Rr must be greater than R1 and R2).
3. From center O2, open a compass to radius (Rr - R2) and draw another arc.
4. The intersection of these two arcs (O3) is the center of the 3.
1. Teacher Activities Introduction (10 minutes): Begin by reviewing basic geometry concepts: circles, radii, perpendicular lines. Introduce the terms "tangent" and "tangency" using simple visual examples (e.g., a wheel touching the ground, a ball resting against a wall). Briefly discuss the relevance of tangency in everyday objects and engineering designs within Nigeria (e.g., curves on roads, parts of local machinery, architectural features). State the learning objectives for the lesson. Concept Explanation and Demonstration (30 minutes): Clearly define "tangent," "point of tangency," and the three fundamental principles of tangency. Use diagrams on the whiteboard or projector. Demonstrate step-by-step the construction of a tangent to a point on the circumference of a circle. Emphasize accurate use of drawing instruments. Demonstrate the construction of common external tangents for both equal and unequal circles. Explain the reasoning behind each step. Demonstrate the construction of common internal tangents for unequal circles. Show how to construct tangent arcs (external and internal) and arcs tangent to lines and circles. Use large-scale diagrams for clarity during demonstrations.
Guided Practice Facilitation (40 minutes): Distribute drawing sheets and instruments. Assign guided practice questions. Circulate around the classroom, observe students' work, and provide individual or small-group assistance. Correct common errors and reinforce proper techniques (e.g., light construction lines, accurate measurement, sharp pencil usage).
Review and Consolidation (10 minutes): Review key concepts and principles. Address any lingering questions from students. Summarize the practical applications of tangency. 3.
2. Student Activities Active Listening and Observation: Students will pay close attention to the teacher's explanations and demonstrations, taking notes as necessary.
Questioning: Students will ask clarifying questions during the teacher's demonstrations and explanations.
Instrument Preparation: Students will ensure they have all necessary drawing instruments (drawing board, T-square, set squares, compass, divider, pencils (HB, 2H), eraser, drawing paper) ready for practical work.
Step-by-step Construction Practice: Students will follow the teacher's instructions to perform guided constructions on their drawing sheets.
Peer Learning: Students may work in pairs or small groups to discuss steps and assist each other under the teacher's supervision.
Accuracy and Neatness: Students will focus on executing constructions accurately and neatly, using appropriate line types (construction lines, visible lines).
Problem-Solving: Students will attempt independent practice questions, applying the learned construction methods. ---
Architecture and Urban Planning (e.g., Abuja City Gates, curved buildings in Lagos): Tangency principles are fundamental in designing aesthetically pleasing and structurally sound curved elements in buildings, bridges, and urban infrastructure. Examples include smoothly connecting curved walls to straight sections, designing arches, and creating elegant transitions in public spaces or recreational areas like parks and stadiums. In road design, tangential curves are critical for creating safe and comfortable transitions between straight sections, preventing sharp turns that could lead to accidents, common in our major expressways. Manufacturing and Engineering Design (e.g., Nnewi industrial clusters, Aba industrial hub): In the production of various machine components such as gears, cams, and pulley systems, tangency ensures smooth operation and efficiency. For instance, the teeth profiles of gears are often based on involute curves, which are essentially derived from tangency concepts, allowing for continuous and uniform power transmission without jarring. Designing tools, automotive parts (e.g., fender curves, headlight shapes), and agricultural machinery also heavily relies on precise tangential constructions to achieve functional and ergonomic shapes. Fashion, Art, and Craft (e.g., Adire fabric patterns, calabash carving): Many traditional Nigerian art forms and modern design practices incorporate intricate geometric patterns and flowing curves. Tangency plays a subtle yet significant role in creating these designs, ensuring that various elements blend seamlessly. For instance, in calabash carving, the lines and circular patterns often meet tangentially to create a harmonious and continuous flow. Similarly, furniture design, textile patterns (e.g., batik or adire), and jewelry often feature elements where curves meet lines or other curves with perfect tangency for visual appeal. ---