Chapter 5 test
equidistant to each side
Perpendicular distances
Orthocenter and altitudes
A triangle has 3 altitudes that are concurrent and cross at the orthocenter, the orthocenter does not need to be inside the triangle like the altitudes.
From the word incenter you should know
Angle bisectors are created when they hit the corner (vertices) Perpendicular distances to the midpoint's of the sides are the same All of the side distances are congruent Can't have a negative length
Incenter Summary
Angle bisectors are not congruent. Midpoint of sides are congruent. The Incenter is the center of the inscribed circle. The sides are equidistant to the incenter. The vertices are angle bisects.
inscribed circle
Because the incenter is the center of the circle and is equidistant to all the sides, the circle is the biggest circle possible that fits inside the triangle and touches all the sides that become radii.
The relation between circumcenter and circle
Circles are set to be circumscribed around the triangles. When you circumscribe a circle it is the smallest circle that hits all of the vertices. The circle with the center of the circumcenter is said to be circumscribed about the triangle. The triangle is inscribed inside the circle.
From the word circumcenter in a problem you should know what?
Everything that looks like a perpendicular bisector is, and that they make right angles. The midpoints of sides. The circumcenter is the same distance from every vertex (cornner)
similar
Figures that have the same shape but not necessarily the same size
Comparing triangles
IF you have 2 sides of one triangle and they are congruent to two other sides of another triangle then you can use the included angle to get information about the 3 side.
Converse of the Angle Bisector Theorem
If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.
Angle Bisector Theorem
If a point is on the bisector of an angle, then the point is equidistant from the 2 sides of the angle.
Bigger angle - bigger side
If one angle of a triangle s larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
Smallest angle - smallest side
If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.
Property of Isosceles triangle
If this is an isosceles triangle, we can prove that the vertex angle's median, which goes to the midpoints, is the same as the altitude. We can take it a step forward because since angles are congruent that means that it is also an angle bisector, and it is also a perpendicular bisector because it is perpendicular and bisecting.
Hinge Theorem
If two sides of a triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.
Converse of the Hinge Theorem
If two sides of a triangle are congruent to two sides of another triangle, and the third side in the first is longer than the third side in the second triangle, then the included angle measure of the first triangle is greater than the included angle measure in the second triangle.
Triangle inequalities tip:
If you know the two values to find the possible answers subtract them and add them. Greater than difference, less than the sum. Can not include a segment in an answer twice. The theory is not disproven when two triangles come together because you have to compare in one triangle then put them together not compare both triangles at once.
Converse of the Perpendicular Bisector Theorem
In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Perpendicular Bisector Theorem
In a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Isosceles Triangle
In an isosceles triangle, the perpendicular bisector, angle bisector, median, ad altitude form the vertex angle to the base are all the same segment.In an equilateral triangle, this is true for the special segment from any vertex.
Why also the line containing the opposite side.
In an obtuse triangle, the orthocenter is outside the triangle as well as the altitudes so they end up hitting the lines containing the opposite sides.
Acute Orthocenter
Inde the triangle; same with circumcenter
When is it good to use the incenter?
It is good to use the incenter when your concern is getting to sides at equal distances.
Medians and midpoints
Medians hit the midpoints of the opposite side, so connect medians to there midpoints then the centroid will reveal itself.
What do you need to know to prove a triangle has a perpendicular bisector
Need to know that it has perpendicular distances. That the angles correspond to each other angles have to be congruent the segment between the bisector and the side, and the bisector and the other side were bisected by the bisector and are congruent and equal.
Centroid on a diagram
Once you have identified the centroid, then you know all the line going through it are the medians.
Obtuse Orthocenter
Outside the triangle and below the obtuse angle if the angle is pointing down. It is above the angle if the obtuse angle is pointing up. Circumcenter is also outside, but usually across from the obtuse angle.
Circumcenter Summary
Perpendicular bisectors are not congruent. Vertices are congruent. The Circumcenter is the center of the circumscribed circle. The sides are perpendicular bisects. The vertices are equidistant to the circumcenter.
Altitudes
Perpendicular to the line containing the midpoint the midpoint of the opposite side. They cross at the orthocenter, and are inside an acute triangle, on a right triangle, and outside and either above or below the obtuse angle based on the angle's position.
Summary of Angle Bisectors
Properties: Bisects Angles Point of concurrency: Incenter Where is it found on the triangle: Inside for acute, Inside for the right triangle, and inside for obtuse. Special Property: The incenter is equidistant from each side of the triangle.
Summary of Perpendicular Bisectors
Properties: Form right angles, Hit midpoint of the opposite side Point of concurrency: Circumcenter Where is it found on the triangle: Inside for acute, On the hypotenuse for the right triangle, and outside and across from the obtuse angle for obtuse. Special Property: The circumcenter is equidistant from each vertex.
Summary of Altitudes
Properties: Form right angles, Start at the vertex Point of concurrency: Orthocenter Where is it found on the triangle: Inside for acute, On the vertex angle (where the right angle would be) for the right triangle, and outside and below or above the obtuse angle depending on whether the obtuse angle is pointing up or down for obtuse. Special Property: The lines containing the altitudes are concurrent at the orthocenter. Also Altitudes make right angles with sides.
Summary of Medians
Properties: Start at the vertex, and Hit the midpoint of the side Point of concurrency: Centroid Where is it found on the triangle: Inside for acute, Inside for the right triangle, and inside for obtuse. Special property: The centroid is two-thirds of the distance from each vertex to the midpoint of the opposite side.
Tip:
Proving triangles congruent helps with everything including seeing if a segment is multiple of the types.
Important message about angle bisectors
Remember also that the distance from a point to a line is the length of the perpendicular segment from the point to the line.
What is the center of the circumscribed circle.
The center of the circumscribed circle is also the place where the perpendicular bisectors meet, so it is the center of the circumscribed triangle which is why it is called the circumcenter. It is where the perpendicular bisectors cross, and depending on what kind of triangle it is, it falls in different locations.
Center of Balance-centriod
The centroid will always be the center of gravity of the triangle. The circumcenter does not have to be inside the triangle let alone the center of gravity. The incenter is inside the triangle but it doesn't have to be a balancing point.
Circumcenter on an obtuse triangle
The circumcenter falls outside the triangle, and it is the same distance to each corner of the triangle.
Concurrency of Altitudes of a Triangle
The lines containing the altitudes of a triangle are concurrent
Concurrency of Perpendicular Bisectors of a Triangle
The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. All perpendicular bisectors are concurrent.
point of concurrency
The point of intersection of the lines, rays, or segments.
How all types come together
The segment that goes from the vertex to the base is a bisector, a perpendicular bisector, a median, and an altitude in an isosceles triangle. In an equilateral every vertices are a bisector, median, a perpendicular bisector, and an altitude. The point where they cross in an equilateral triangle is a circumcenter, centroid, incenter, and orthocenter always.
The triangle inequality theorem
The sum of any two side lengths of a triangle is greater than the third side length.
Centroid
The three medians of a triangle are concurrent. The point of concurrency called the centroid is inside the triangle. The center of balance.
Need to know
The two longer words circumcenter and and orthocenter are inside, outside, on. And the two shorter, incenter and centroid, are always inside.
Altitudes in depth
They are always concurrent at the orthocenter. In a right triangle the two legs are altitudes. The third would from where the right angle is.
Right Orthocenter
They are congruent on the vertex angle (where the right angle would be). On the triangle. The circumcenter is usually the midpoint of the hypotenuse.
Prove of the short long relation
We can prove the smaller of the two similar triangles is 1/2 of the big triangle, so we can prove the shorter side is 1/2 of the larger.
Median relations
When the centroid breaks the median into two points there is always a short part and a longer part. The relationship is that the shorter part (form the side midpoint to the centroid) is 1/3 of the median, and the longer part (form the vertices to the centroid) is 2/3 of the median. The good thing about 1/3 and 2/3 is that they are in the ratio of 1:2. But the short part is always half the long part. 2(short) = long or 1/2(long) = short.
Concurrency
When three or more lines, rays, or segments intersect in the same point, they are called concurrent lines, rays, or segments.
angle bisector
a ray that divides an angle into 2 congruent angles
median of a triangle
a segment from a vertex to the midpoint of the opposite side
perpendicular bisector
a segment, ray, line, or plane that is perpendicular to a segment at its midpoint
equidistant
equally distant
Relation between length and angle measures
largest angle measure opposite to largest side. Smallest angle measure opposite to smallest side.
right triangle circumcenter
perpendicular bisectors cross on the hypotenuse making the circumcenter on the hypotenuse. Still has the same distance to the vertices.
acute triangle circumcenter
somewhere in the triangle, still same distance to each vertices.
Concurrency of Angle Bisectors of a Triangle
the angle bisector of a triangle intersect at a point that is equidistant from the sides of the triangle
Concurrency of Medians of a Triangle
the medians of a triangle intersect at a point that is 2/3 of the distance from each vertex to the midpoint of the opposite side
Altitudes of a triangle
the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side and creates right angles. Like what you would think of as the height of a triangle when solving for the area.
Orthocenter
the point at which the lines containing the three altitudes of a triangle intersect
Incenter
the point of concurrency of the three angle bisectors of a triangle
Circumcenter
the point of concurrency of the three perpendicular bisectors of a triangle .he circumcenter is equidistant from the three vertices, so the circumcenter is the center of a circle that passes through all three vertices.
