Math Journal 5 Perpendicular Bisector What is a

Math Journal 5

Perpendicular Bisector What is a Perpendicular Bisector? A Perpendicular Bisector a line or ray that passes through the midpoint of the segment Theorem: If a line pass through a point in the segment that are equidistant from the endpoint then is a perpendicular bisector. Converse: If the line that passes though a point in the segment and its not equidistant then is not a perpendicular bisector.

Examples A 1) If AB=25. 4 cm AD=20 cm and DB=12 cm find AB. AB= 25. 4 2) If H is perpendicular bisector of CB and DB is 4. Find CB. CB= 8 3) Given that H is perpendicular bisector of segment CB and AB is 4 x+4 and AC is 8 x-6. Find AB AB=AC C D H 4 x+6=8 x-6 -4 x +6=4 x-6 +6 +6 4 x=12 X=3 AB=4(3)+4 AB=16 B

Angle Bisector What is an Angle Bisector? A Ray or line that divides the angle is two congruent angle Theorem: If a point is in the line of the angle bisector then is equidistant from the sides of the angles Converse: If its equidistant from the sides of the angles then it on the angle bisector

Examples 1) <ABD congruent <ACD 2) <DAC congruent <DAB 3) <BDA congruent <CDA A B D C Counter: 1) Since AB is congruent to AB, AD is an angle bisector of triangle BAC 2) Since BD is congruent to DC, AD is an angle bisector of triangle BAC

Concurrent What is concurrent: Concurrent is when three or more lines meet together making a point.

the concurrency of Perpendicular bisectors of a triangle theorem The Perpendicular bisector of a triangle is a point where its equidistant from the vertices of the triangle

Circumcenter is where the point of the 3 perpendicular bisector meets, its special because the point is equidistant through all the vertices of the triangle.

the concurrency of angle bisectors of a triangle theorem When a point is made and is equidistant from the sides of the triangles.

A point made by the meeting of the three angle bisector, and it equidistant from the sides of the triangle. INCENTER Incenter

The median of a triangle is a segment that goes from the vertex to the opposite side midsegment Median

A point where all the median meets. The distant from the centroid to the vertex will the double from the distant from the centroid to the side. Centroid

the concurrency of medians of a triangle The median of a triangle that theorem intersect a point creating a line that is double the other part of the line 2 1. 75 3. 3 2. 5 2 3 5 6. 6 4 1 3. 5 0. 4 1. 43 6. 2 3. 1 0. 2 2. 86 6

altitude of a triangle Segment that is perpendicular to the other angles.

Orthocenter The point that all altitude lines meet of a triangle.

The concurrency of altitudes of a triangle theorem A point that the altitudes intersects in a triangle.

What is Midsegment? Is a segment joined by the midpoint of the sides, the triangle has 3 midsegment in total. Theorem: The midsegment is parallel to the opposite side and it’s half the size of the side. Midsegment

*ATTENTION* This Images are not draw in scale : P EXAMPLES In the triangle they are three segments that are related to the angle, meaning that if one segment is the biggest then to the opposite side angle will be the biggest one and if the segment is the shortest one the opposite side angle will be the shortest one. the relationship between the longer and shorter sides of a triangle and their opposite angles

The angle that is an extension of a interior angle. A+B= the exterior angle b c a A+B b b a a c A+B the exterior angle inequality c A+B

Triangle Inequality 1. 2, 5, 8 No, because adding 2+5<8 and this will not make a triangle. 2. 15, 28, 40 Yes because adding 15+28>40 and this will make a triangle 3. 16, 24, 40 No because adding 16+24=40 making two line with the same lengths. The sum of 2 sides will be greater than the 3 rd side.

Indirect Proof It’s a way to proof not directly but indirectly. STEPS TO SOLVE THIS 1)Assume the Given is false 2) Use that as a give and start the proof 3)When you come to a contradiction that will be your last step.

Examples Prove A triangle cannot have 2 right angles Statement Reason Assume a triangle has 2 right <1=<2 Given M<1=m<2=90 degrees Def of Right Angle M<1+M<2=180 degrees Substitution M<1+m<2+m<3=180 Triangle sum thrm M<3=0 contradiction A triangle cannot have 2 Right Angles

Proof: two supplementary angles cannot both be obtuse angles Statement Reason Two supplementary angles can both be obtuse angles Given X>90 degrees Def of obtuse angles (X>90)+(x>90) Substitution (x>90)+(x>90)=180 contradiction Two supplementary angles cannot both be obtuse.

A Triangle cannot have 2 obtuse angles Statement Reason A triangle can have 2 obtuse angles Given X>90 Def of obtuse angles M<1+m<2+m<3=180 Def of a triangle X can be 91 or more Def of obtuse angle 100+m<3=180 Substitution 200+M<3=180 Addition M<3=-20 Subtraction M<3=-20 contradiction Triangle cannot have 2 obtuse angles

Hinge Theorem Two triangles with two congruent segments but the 3 rd segment will different and will be larger if the angle that is in the opposite side will be bigger.

the special relationships in the special right triangles (30 -60 -90 and 45 -4590) 45 -45 -90: In this triangle the legs will the same size but the hypotenuse will be length times √ 2 30 -60 -90: In this triangle the base will the smallest size the longest side would be the hypotenuse that is the double of the base angle and the height will be the base time the √ 3

Examples 45 1 X would be 1√ 2 because you multiply the legs time √ 2 x 45 1 X would be 8√ 2 because we need to divide 16 by √ 2, and we know that we can’t have the √ 2 as a denominator so we multiply everything by √ 2 and later we have 16 √ 2 divided by 2 and we can simplify by 2 and the answer would be 8 √ 2 45 16 X 45

45 x 7 X would be 7 √ 2/2 30 45 x X is going to be 8 Y will be 4 √ 3 y 60 4

30 X= 9 because the hypotenuse is the double of the base of the triangle Y= is going to be 9 √ 3 because we already know that the base is 9 18 y 60 x 30 y X=3 √ 3 Y=6 √ 3 because to find the shortest side we need to dived 9 by √ 3 that is 3 √ 3 making X and we multiply time 60 2 the 3 √ 3 x 9