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# Ching chi tu journal 5

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• 1. Ching Chi Tu Journal#5 m1 geometry
• 2. Perpendicular Bisector A line perpendicular to a segment at the segment’s midpoint. Perpendicular Bisector Theorem – If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Converse of the Perpendicular bisector Theorem – If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
• 3. Converse P.B. Thm H A Ex 1. f = g Ex 2. d = e Ex 3. b = c T T f = g; g = c; i = h ce ab; bg cb Ex 1. a = h Ex 2. i = j Ex 3. cg = bg T
• 4. Angle Bisector Angle Bisector - A ray that divides an angle into two congruent angles. Angle Bisector Theorem – If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Converse of Angle Bisector Theorem – If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle
• 5. Ex 1. c bisects <abc; point L is equidistant from both sides. Ex 2. i bisects <ghi; point J is equidistant from both sides. Ex 3. e bisects <def; point K is equidistant from both sides. H A.B. Thm Converse Ex 1. L is equidistant from both sides; c must be an a.b. Ex 2. J is equidistant from both sides; i must be an a.b. Ex 3. K is equidistant from both sides; f must be an a.b.
• 6. Concurrency To be concurrent means for three or more lines to intersect at one point. The point of concurrency of perpendicular bisectors form a circumcenter. Circumcenter - where three perpendicular bisectors of a triangle are concurrent. It is equidistant from the vertices of the triangle
• 7. Circumcenter D is the circumcenter of triangle abc, it is equidistant from points a, b, and c. Right = circumcenter midpoint of hypotenuse
• 8. Circumcenter L is the circumcenter of triangle jkl, it is equidistant from points j, k, and l. Acute = circumcenter inside
• 9. Circumcenter H is the circumcenter of triangle efg, it is equidistant from points e, f, and g. Obtuse = circumcenter outside
• 10. Concurrency The point of concurrency of angle bisectors forms an incenter. Incenter – The point of concurrency of three angle bisectors of a triangle. It is equidistant from the sides of the triangle. Concurrency Concurrency
• 11. Incenter Incenter always inside triangle D is the incenter of triangle abc, it is equidistant to all sides.
• 12. Incenter Incenter always inside triangle A is the incenter of triangle xyz, it is equidistant to all sides.
• 13. Incenter Incenter always inside triangle D is the incenter of triangle abc, it is equidistant to all sides.
• 14. Concurrency Median – segments whose endpoints are a vertex of the triangle and the midpoint of the opposite side. The point of concurrency of medians is called a centroid. Centroid – Point of concurrency of the medians of a triangle. It is located 2/3 of the distance from each vertex to the midpoint of the opposite side
• 15. Centroid Ex 1. g is the centroid of triangle abc Ex 2. o is the centroid of triangle egc Ex 3. k is the centroid of triangle adg Centroid = Point of balance
• 16. Concurrency Altitude – a perpendicular segment from a vertex to the line containing the opposite side. The point of concurrency of three altitudes of a triangle creates an orthocenter. Orthocenter – point where three altitudes of a triangle are concurrent. Nothing special.
• 17. Orthocenter D is the orthocenter of triangle abc with lines e, d, and f as altitudes.
• 18. Orthocenter D is the orthocenter of triangle abc with lines e, d, and f as altitudes.
• 19. Orthocenter D is the orthocenter of triangle abc with lines e, d, and f as altitudes.
• 20. Midsegment Midsegment of a triangle – a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, forming a midsegment triangle. Triangle Midsegment Theorem – A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side.
• 21. Midsegment Ex 1. f  BC Ex 2. d  AC Ex 3. e  BA They are all midsegments and are parallel to the lines written beside them.
• 22. Relationships In any triangle, the longest side is always opposite to the largest angle. Small side = small angle.
• 23.
• Ex 3.
• <a is the smalles angle, so its opposite angle is also the smallest.
• Side b is the largest side so its opposite angle is the largest.
• Ex 2.
• <c is the largest angle so its opposite side is also the largest.
• Side a is the smallest side so its opposite angle is also the smallest.
• Ex. 1
• <a is the largest angle so therefore side a, its opposite side, must be the largest side.
• <c has the smallest angle so this means that side c is the smallest side in the triangle.
• 24. Inequality Exterior Angle Inequality – The exterior angle of a triangle is larger than either of the non-adjacent angles.
• 25. Ex 1. m<e is larger than all non adjacent angles such as b and y. Ex 2. m<c is greater than all non adjacent angles such as a and y. Ex 3. m<o is greater than all non adjacent angles such as a and b.
• 26. Inequality Triangle Inequality Theorem – The two smaller sides of a triangle must add up to MORE than the length of the 3 rd side.
• 27. Ex 1. 2.08 + 1.71 is less than 5.86 so it doesn’t form a triangle. Ex 2. 3.02 + 2.98 = 6 But it doesn’t form a triangle, it forms a line/segment. Ex 2. 1.41 + 1.41 is greater than 2 so therefore, it does form a triangle.
• 28. Indirect Proofs
• Steps for indirect proofs:
• Assume the opposite of the conclusion is true