Angle Bisector & Perpendicular Bisector.avi
Summary
TLDRThis educational video delves into the concepts of angle bisectors and perpendicular bisectors in the context of triangles. It explains that an angle bisector divides a vertex angle into two equal parts, meeting at the incenter, while a perpendicular bisector cuts a side in half at a 90-degree angle, intersecting at the circumcenter. The video also includes practice problems to illustrate the application of these geometric principles, emphasizing the congruence of angles and segments in solving for unknown variables.
Takeaways
- đ An angle bisector is a line segment that bisects one of the vertex angles of a triangle, creating two equal angles.
- đ Angle bisectors can be found in three different locations within a triangle, and they all intersect at a common point called the incenter.
- đ The incenter is a point of concurrency where the three angle bisectors of a triangle meet.
- đ A perpendicular bisector is a line that is both perpendicular to a side of a triangle and bisects it, passing through its midpoint.
- âïž Perpendicular bisectors have the properties of being at a 90-degree angle and bisecting a segment into two equal lengths.
- đ The circumcenter is the point where the three perpendicular bisectors of a triangle intersect, and it can be inside or outside the triangle.
- đą To find the measure of an angle bisected by a line, set the expressions for the two resulting angles equal to each other and solve for the variable.
- đ The sum of the two angles created by an angle bisector is equal to the original angle of the triangle.
- đ Each triangle can have three perpendicular bisectors, each corresponding to a different side of the triangle.
- đ The circumcenter is named for the fact that it is equidistant from all vertices of the triangle, which is a key property in circle geometry.
- đą Solving for the length of a segment bisected by a perpendicular bisector involves using the given angle measure and the properties of right triangles.
Q & A
What is an angle bisector in the context of a triangle?
-An angle bisector is a line segment that bisects one of the vertex angles of a triangle, dividing it into two equal angles.
How many angle bisectors can a triangle have?
-A triangle can have three angle bisectors, one for each of its angles.
What is the point of concurrency formed by the intersection of all three angle bisectors in a triangle called?
-The point of concurrency formed by the intersection of all three angle bisectors is called the incenter.
What is the definition of a perpendicular bisector?
-A perpendicular bisector is a line segment that is both perpendicular to a side of a triangle and bisects it, passing through its midpoint.
How does a perpendicular bisector relate to the concepts of a median and an altitude?
-A perpendicular bisector combines properties of both a median and an altitude: it goes through the midpoint of a side (like a median) and forms a 90-degree angle (like an altitude).
What is the point of concurrency for the perpendicular bisectors of a triangle known as?
-The point of concurrency for the perpendicular bisectors of a triangle is called the circumcenter.
Can the circumcenter of a triangle always be found inside the triangle?
-No, the circumcenter can be either inside or outside the triangle, depending on the triangle's shape.
In the practice problem involving an angle bisector, how do we determine the value of x if angle 1 is 6x - 10 and angle 2 is 4x + 12?
-Since the angle bisector divides the angle into two equal parts, we set 6x - 10 equal to 4x + 12, solve for x, and find that x equals 11.
What is the measure of angle LMG if it is bisected by the line segment LM?
-If angle 1 and angle 2 are both 56 degrees, then angle LMG, which is the sum of angle 1 and angle 2, is 112 degrees.
In the second practice problem, how do we find the length of side BC if the perpendicular bisector ED intersects BC at D and BD is 2x + 4?
-First, we solve for x using the equation 7x + 6 = 90, finding x to be 12. Then, we calculate BD as 28 and since D is the midpoint, DC is also 28, making BC equal to 56.
What is the significance of the 90-degree angle in the definition of a perpendicular bisector?
-The 90-degree angle in the definition of a perpendicular bisector signifies that the line is perpendicular to the side of the triangle it bisects.
Outlines
đ Understanding Angle Bisectors and Perpendicular Bisectors
This paragraph introduces the concepts of angle bisectors and perpendicular bisectors in the context of triangles. An angle bisector is a line segment that divides a vertex angle of a triangle into two equal parts. For instance, in triangle ABC, a line segment BD is drawn to bisect angle B, making angles ABD and DBC equal. The paragraph explains that each triangle can have three angle bisectors, all of which intersect at a common point known as the incenter. The concept of a perpendicular bisector is also discussed, which is a line segment that bisects a side of a triangle and forms a 90-degree angle with it. The midpoint of the side is bisected, and the perpendicular bisector combines properties of a median and an altitude. Three perpendicular bisectors can be drawn in a triangle, intersecting at a point called the circumcenter, which can be inside or outside the triangle.
đ Solving Problems Involving Angle and Perpendicular Bisectors
The second paragraph focuses on solving practice problems related to angle bisectors and perpendicular bisectors. The first problem involves an angle bisector that bisects an angle into two equal parts. Given two expressions for the angles, the paragraph demonstrates how to set up an equation to find the value of 'x' that makes the angles congruent. The solution process involves algebraic manipulation to solve for 'x' and then uses this value to determine the measure of the bisected angle. The second problem deals with a perpendicular bisector, which bisects a side of a triangle and forms a right angle. The paragraph shows how to use the given information to set up an equation and solve for the length of the bisected side. The process includes using the properties of perpendicular bisectors to find the lengths of the segments and ultimately the full length of the side.
Mindmap
Keywords
đĄAngle Bisector
đĄPerpendicular Bisector
đĄTriangle
đĄVertex Angle
đĄIncenter
đĄMidpoint
đĄCongruence
đĄCircumcenter
đĄAltitude
đĄMedian
đĄCongruency Postulate
Highlights
Introduction to the concept of an angle bisector as a line segment that bisects one of the vertex angles of a triangle.
Explanation of the term 'bisect' as cutting something into two equal sections, with an example of angle ABD and angle DBC being equal.
Mention of the existence of three angle bisectors in a triangle, all intersecting at a common point called the in-center.
Differentiation between an angle bisector and a perpendicular bisector, with the latter involving a 90-degree angle and cutting a segment in half.
Description of the perpendicular bisector's properties, including forming a 90-degree angle and passing through the midpoint of a side.
Illustration of the three perpendicular bisectors in a triangle and their intersection at the circumcenter, a point of concurrency.
Note on the circumcenter's variability in location, being sometimes inside or outside the triangle.
Practice problem involving an angle bisector where the angles are expressed in terms of x, leading to an equation to solve for x.
Solution of the practice problem by setting up an equation based on the congruence of bisected angles and solving for x.
Calculation of the measure of angle LMG by adding the measures of the bisected angles, demonstrating the angle bisector theorem.
Introduction of a second practice problem involving a perpendicular bisector and the properties of a right angle and midpoint.
Setting up an equation to solve for x based on the given angle measure of 90 degrees and the expression 7x plus 6.
Solving for x and subsequently calculating the lengths of segments BD and DC using the value of x.
Determination of the total length of segment BC by adding the lengths of BD and DC, showcasing the application of the perpendicular bisector.
Emphasis on the practical applications of angle and perpendicular bisectors in solving geometric problems.
Summary of the importance of understanding the properties and applications of angle bisectors and perpendicular bisectors in geometry.
Transcripts
okay class in our video today we're
going to talk about angle bisector and
perpendicular bisector an angle bisector
first of all is a line segment and it is
a line segment that bisects one of the
vertex angles of a triangle so here we
have triangle ABC and we're going to
draw a line segment that is going to
bisect angle B so here what we will call
BD is our angle bisector now if you
remember bisect means to cut something
into two pieces to two equal sections so
angle abd is going to be the same as
angle DBC and then if we also think
about our definition of a bisector angle
abd plus angle DBC when I put them
together is going to give me the whole
thing angle a B C so that right there is
our definition of an angle bisector now
just like with median and with altitude
we can have three angle bisectors in
each triangle so for example here we
have eg this is an angle bisector here
we have D H this is also an angle
bisector and here we have F J and this
is also going to be an angle bisector
okay so here we have three different
angle bisectors and notice once again
they all meet or they intersect at a
common point another one of our points
of concurrency and this time we call it
an end Center so remember with median
when all three medians intersected we
had a centroid when all three altitudes
intersected we had an ortho Center and
now when all three angle bisectors
intersect we have what's called an in
center moving on now we're going to talk
about a perpendicular bisector and a
perpendicular bisector does two things
first of all it's a line segment as it
definitely beginning and definite in
perpendicular tells me that it's going
to form a 90 degree angle and bisector
tells me that it's going to cut
something in half now as opposed to an
angle bisector the perpendicular
bisector is going to cut a segment in
half as opposed to an angle so if we're
cutting a segment in half it's going to
go through the midpoint of one of these
sides so here we have our perpendicular
bisector the perpendicular part tells me
it's going to be 90 degrees and the
bisector part tells me that it's going
through the midpoint so if M is my
midpoint that means this piece is equal
to this piece so not only is it
bisecting St in half it's also forming a
90 degree angle so this is what we call
our perpendicular bisector in this case
it is in P okay
so M is the midpoint SM is congruent to
Mt SN plus MT is going to equal the
whole thing s T and that's that segment
addition postulate and because it's
perpendicular we say SNP is 90 degrees
this is kind of a combination of a
median and an altitude remember the
median goes to the midpoint and the
altitude forms a 90 degree angle so
while this these are not this is not
both an altitude in the median it's kind
of a combination of the two it has those
two properties 90 degrees and it goes
through the midpoint and just like with
our three previous special segments we
can have three perpendicular bisectors
in one triangle so our first one we'll
call it our s because s is the midpoint
of BC and it forms a 90-degree angle
we also will have BD because this is the
midpoint of AC and it forms a 90 degree
angle and lastly we also have L M
because M is the midpoint of a B and it
forms a 90 degree angle so all three of
those are s B D and M L are
perpendicular bisectors now notice they
also intersect at a common point and
like before we have a special name for
that we call this the circumcenter so
remember median was centroid altitude
was orthocenter angle bisectors form in
in center and perpendicular bisectors
form a circumcenter okay now if you
notice this time the point of
concurrency or the circumcenter is on
the outside of the triangle that will
happen sometimes with circum centers
whereas our others were in the truck
inside the triangle the circumcenter can
be both inside sometimes or outside it
just depends
so now let's do some practice problems
our first practice problem says that km
is an angle bisector which means it is
cutting this angle into two equal parts
and so if angle one is 6x minus 10 and
angle 2 is 4x plus 12 we know that the
two pieces have to be congruent that's
what bisect means cut in half so now we
have our equation 6x minus 10 equals 4x
plus 12 and we just need to solve our
and subtract 4x from both sides and that
gives us 2x minus 10 equals 12 then
we're going to add 10 to both sides and
when we simplify we get 2x equals 22 the
last step obviously is to divide by 2
and so our final answer is x equals 11
but it wants us to figure out what is
the measure of angle L in G well if you
look L M G is a combination of angle 1
and angle 2 so we're going to go find
the angle 1 when we plug in is 6 times
11 minus 10 which gives me 56 degrees an
angle 2 when we plug in is 4 times 11 44
plus 12 also gives me 56 degrees which
is what we said if it's an angle
bisector these two angles have to be
congruent which is what we proved here
but to get the final product LMG we have
to add angle 1 plus angle 2 so we're
going to do 56 degrees plus 56 degrees
gives me 112 degrees so our final answer
is 112 now we're going to move on to our
next question so here we have our second
practice problem and involves a
perpendicular bisector we have triangle
ABC where edie is my perpendicular
bisector and remember a perpendicular
bisector does two things
perpendicular tells me that that angle
is 90 degrees bisector tells me that
we're cutting these the segment BC and
so D is the midpoint which means this
segment is going to be congruent to this
segment all right now if that's the case
and we say that BD is 2 X plus 4 in
order to set anything up we need to know
what this side is but we don't know that
yet the one thing we do know is that
angle EDC which is 90 degrees is
supposed to be 7x plus 6 since we do
know that for sure I can set my equation
up as 7x plus 6 equals 90 degrees and I
can solve from here subtract 6 from both
sides simplify and I get 7x equals 84
then I'm going to divide both sides by 7
and our final answer is x equals 12 but
again we want to find the measure of BC
well now that I know what X is I can go
plug it in and so I'm going to plug in
well BD is going to be 2 times 12 which
is 24 plus 4 is 28 and if this side is
congruent to this side then DC is also
going to be 28 and if I'm looking for
the whole thing BC I'm going to take
this piece 28 and add it to this piece
28 so BD plus DC is going to give me BC
so the whole thing is 28 plus 28 which
equals 56
you
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