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Dr. Mohamed BEN ALI

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By the end of this lecture, students will be able to: Understand the types of Tangents. Construct tangents. Construct incircle. Construct circumcircle Objectives

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Centre - the point within the circle where the distance to points on the circumference is the same. radius - the distance from the centre to any point on the circle. The diameter is twice the radius. circumference(perimeter) - the distance around a circle. diameter - a chord(of max. length) passing through the centre Some definitions

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chord is a straight line joining two points on the circumference. If line intersect the circle at two point that is called secant tangent - a straight line making contact at one point on the circumference, such that the radius from the centre is at right angles to the line. Some definitions

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Given a circle, and any point on it, we can draw a straight line through the point that will be tangent to this circle

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Step 1: Draw the line through C and T C T

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Step 2: Draw a circle about T that passes through C, and let D denote the other end of that circle’s diameter C T D

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Step 3: Construct the straight line which is the perpendicular bisector of segment CD C T D tangent-line

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Drawing a circle tangent to a line at a given point At P, draw a perpendicular to the line AB Set off the radius of the required circle on the perpendicular Draw a circle with radius CP

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Drawing tangents to two circles Move the triangle and T-square as a unit until one side of the triangle is tangent, by inspection, to the two cirles, Then slide the triangle until the other side passes through the centre of one circle, and lightly mark the point of tangency, Then slide the triangle until the side passes through the centre of the other circle, and mark the point of tangency. Finally, slide the triangle back to the tangent lines between the two points of tangency. Draw the second tangent line in a similar manner

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Drawing two tangents circles Internally Tangent Externally Tangent

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Circle 1: centre C and radius R1 Circle 2: centre C’ and radius R2 Step 1: Draw the line through C and T Step 2: Set your compass for radius R2 Step 3: Set your pointer on T. Step 4: Make a mark on the right side of the line and label it C’ Step 5: Set your pointer on C’ and draw the circle 2. How do we do it? External tangent C T C’

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Circle 1: centre C and radius R1 Circle 2: centre C’ and radius R2 Step 1: Draw the line through C and T Step 2: Set your compass for radius R2 Step 3: Set your pointer on T. Step 4: Make a mark on the left side of the line and label it C’ Step 5: Set your pointer on C’ and draw the circle 2. How do we do it? Internal tangent C T C’

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Drawing an arc tangent to a line or arc and through a point

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Drawing an Arc Tangent to a Line and an Arc Given line AB and arc CD AB C D Strike arcs R 1 (given radius) R1R1 R1R1 Draw construction arc parallel to given arc, with center O O Draw construction line parallel to given line AB From intersection E, draw EO to get tangent point T 1, and drop perpendicular to given line to get point of tangency T 2 E T1T1 T2T2 Draw tangent arc R from T 1 to T 2 with center E

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Drawing an arc tangent to two lines at Right Angles Given two lines AB and BC with right angle ABC With D and E as the points, strike arcs R equal to given radius A B C R R R With B as the point, strike arc R equal to given radius O E D With O as the point, strike arc R equal to given radius

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Drawing an Arc Tangent to Two Lines at an Acute Angle A B C D Given lines AB and CD Draw parallel lines at distance R Draw the perpendiculars to locate points of tangency With O as the point, construct the tangent arc using distance R R R O

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C D Given lines AB and CD Construct parallel lines at distance R Construct the perpendiculars to locate points of tangency With O as the point, construct the tangent arc using distance R R A B R O

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Drawing an Arc Tangent to Two Arcs Given arc AB with center O and arc CD with center S S D C O B A Strike arcs R1 = radius R R1R1 R1R1 Draw construction arcs parallel to given arcs, using centers O and S Join E to O and E to S to get tangent points T E T T Draw tangent arc R from T to T, with center E R

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Given an ellipse, and any point on it, we can draw a straight line through the point that will be tangent to this ellipse F1F2

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Step 1: Draw a line through the point T and through one of the two foci, say F1 F1F2 T

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Step 2: Draw a circle about T that passes through F2, and let D denote the other end of that circle’s diameter F1F2 T D

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Step 3: Locate the midpoint M of the line- segment joining F2 and D F1F2 T D M

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Step 4: Construct the line through M and T (it will be the ellipse’s tangent-line at T) F1F2 T D M tangent-line

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Observe that line MT is the perpendicular bisector of segment DF2 (because ΔTDF2 will be an isosceles triangle) F1F2 T D M tangent-line

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A B C o Construct a Δ ABC Bisect the side AB Bisect the side BC The two lines meet at O From O Join B Taking OB as radius draw a circumcircle.

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A B C Construct a Δ ABC The two lines meet at O Taking OP as radius Draw a circumcircle Bisect the ABC Bisect the BAC Taking O draw OP AB O P

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