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Journal Chapter 5 . Jose Antonio Pomes . Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse. Give 3 examples of each. A perpendicular bisector is when a line crosses another one forming a 90 degree angle between them.
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Journal Chapter 5 Jose Antonio Pomes
Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse. Give 3 examples of each. • A perpendicular bisector is when a line crosses another one forming a 90 degree angle between them. • The perpendicular bisector theorem stats that if a point lies on the perpendicular bisector, then it is equidistant from both of the endpoints of the segment. • Its converse stats that if a point lies on the perpendicular bisector of a segment if and only if it is equidistant from both endpoints of the segments.
Some examples are… 1. 2. 3.
Describe what an angle bisector is. Explain the angle bisector theorem and its converse. Give at least 3 examples of each. • An angle bisector is when a line cuts an angle exactly in half. • The angle bisector theorem states that if a point lies on the interior of an angle and is on the angle bisector, then its equidistant to both sides of the angle. • Its converse states that if a point lies on the interior of an angle and is on the angle bisector if and only if its equidistant to both sides of the angle.
Some examples are… 2. 1. 3.
Describe what concurrent means. Explain the concurrency of Perpendicular bisectors of a triangle theorem. Explain what a circumcenter is. Give at least 3 examples of each. • Concurrent means that its happening at the same time. • The concurrency of perpendicular bisectors of a triangle theorem stats that the perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. • Circumcenter is the point where the three perpendicular bisectors of a triangle meet.
Describe the concurrency of angle bisectors of a triangle theorem. Explain what an incenter is. Give at least 3 examples of each. • The concurrency of an angle bisector of a triangle theorem stats that the angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. • An incenter is a point of congruency of the angle bisectors of a triangle. It is also equidistant from the sides of the triangles.
Describe what a median is. Explain what a centroid is. Explain the concurrency of medians of a triangle theorem. Give at least 3 examples of each. • A median is a line drawn from the vertex of a triangle to the midpoint of the opposite side. • The centroid is the point that connects the lines drawn from the midpoint to the vertex of the triangle. • The concurrency of medians of a triangle stats that the medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.
Describe what an altitude of a triangle is. Explain what an orthocenter is. Explain the concurrency of altitudes of a triangle theorem. Give at least 3 examples. • -An Altitude is a straight line that goes thru the vertex and perpendicular to the opposite side of it. This forms a 90 degree angle. • An orthocenter is where the three altitudes of a triangle meet or intersect. • The concurrency of altitudes of a triangle theorems stats that the lines containing the altitudes of a triangle are concurrent.
Describe what a midsegment is. Explain the midsegment theorem. Give at least 3 examples. • The midsegment is just a line that connects any two midpoints in a triangle. • The midsegment theorem stats that the segment joining the midpoints of two sides of a triangle is parallel to the third side, and is half the length of the third side.
Describe the relationship between the longer and shorter sides of a triangle and their opposite angles. Give at least 3 examples. • The relationship between the longer and shorter sides of a triangle are that the two small sides have to add up to more than the longer side. If this is not true then the shape cannot be a triangle because its impossible for the segments to connect. • In any triangle the side opposite the biggest angle will always have the longest length. The side opposite the smaller angle will have the smallest length.
Some examples are... 6 8 6 12 5 8 12 10 20
Describe the exterior angle inequality. Give at least 3 examples. - The exterior angle inequality states that if the base of the triangle continues the other segment to form another triangle then that new exterior angle should be bigger that the other two angles that are non-adjacent.
Some examples are... 55 55 120 135 90 30 30 30 150
Describe the triangle inequality. Give at least 3 examples. - The triangle inequality states that in any triangle any two sides must add up to more than the 3rd side.
Some examples are... 7 7 15 9 10 9 17 17 30
Describe how to write an indirect proof. Give at least 3 examples. - To write an indirect proof first you need to assume its false then you solve normally with your theorems and you will find out that you cant prove it and instead you will prove something else.
Some examples are... Marie,a lawyer, was defending her client, CP Rail, in a lawsuit launched by Jeff Miller over alleged injuries incurred in a train accident outside Toronto.Miller claimed that while he was sleeping in his berth, the train had slammed on its brakes while backing up causing his head to strike the wall behind his pillow.He claimed he had sustained a skull fracture causing him great stress on the job and in his personal life, due to headaches.Miller was suing CP Rail for two million dollars.Marie presented the defense's case saying," Assume Mr. Miller is telling the truth. On the night when the accident occurred, the porter made up all the berths in the railway car with the heads toward the front of the train. The train was backing up when it came to its sudden stop. The force of the braking would cause Mr. Miller's feet to strike the wall, not his head.Therefore, Mr.Miller is not telling the truth."CP Rail won the case. Sarah left her house at 9:30 AM and arrived at her aunts house 80 miles away at 10:30 AM. Use an indirect proof to show that Sarah exceeded the 55 mph speed limit.SolutionSuppose that the given statement is false. That is: Sarah did NOT exceed the 55 mph speed limit.She drove 80 miles at 55 mph. At this speed, Sarah would need 80/55 (approximately) = 1 hour 27 minutes to reach her aunts place.But as per the problem she drove from 9:30 AM to 10:30 AM exactly an hour.SO, she must have driven faster than 55 mph.a contradiction to our assumption that Sarah did NOT exceed the speed limit.Therefore, Sarah exceeded the speed limit. a court room trying to prove/disprove someone was guilty of a crime. The "proof by contradiction" could go something like: a person is accused of committing a crime. You are his lawyer, trying to prove he is innocent. Well, what if he DID commit the crime? Well, say the crime occurred in LA. But, we have records that show he was on a flight to Paris at the same time the crime occurred. Since a person cannot be in two places at the same time, we have a contradiction!
Describe the hinge theorem and its converse. Give at least 3 examples. • The hinge theorem states that if two sides of two different triangles are congruent and the angle between them is not congruent then the triangle with the larger angle will have the longer third side. • The converse of the hinge theorem states that if the two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is greater than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.
Some examples are... • A door because the more open you have it the longer the segment and the smaller the angles. • If you have a hard case for your glasses its probably a hinge examples because it also works like the door. • Also a bracelet that it works the same way.
