Angle Bisector & Perpendicular Bisector.avi

Summary

This 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

• 📐 An angle bisector is a line segment that bisects one of the vertex angles of a triangle, creating two equal angles.
• 📏 Angle bisectors can be found in three different locations within a triangle, and they all intersect at a common point called the incenter.
• 📍 The incenter is a point of concurrency where the three angle bisectors of a triangle meet.
• 🔍 A perpendicular bisector is a line that is both perpendicular to a side of a triangle and bisects it, passing through its midpoint.
• ⚒️ Perpendicular bisectors have the properties of being at a 90-degree angle and bisecting a segment into two equal lengths.
• 📍 The circumcenter is the point where the three perpendicular bisectors of a triangle intersect, and it can be inside or outside the triangle.
• 🔢 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.
• 📐 The sum of the two angles created by an angle bisector is equal to the original angle of the triangle.
• 📏 Each triangle can have three perpendicular bisectors, each corresponding to a different side of the triangle.
• 📐 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.
• 🔢 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

00:00

📐 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.

05:01

🔍 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.

Keywords

💡Angle Bisector

An angle bisector is a line segment that divides one of the vertex angles of a triangle into two equal parts. In the video, it is used to define the bisector of angle B in triangle ABC as BD, where angle ABD is equal to angle DBC. This concept is central to understanding the properties of triangles and their symmetry.

💡Perpendicular Bisector

A perpendicular bisector is a line segment that is both perpendicular to a side of a triangle and bisects that side into two equal lengths. The script explains that it cuts through the midpoint of a side, forming a 90-degree angle, as seen with the perpendicular bisector in triangle ABC where it bisects side BC at point M.

💡Triangle

A triangle is a polygon with three edges and three vertices. The video uses triangle ABC to illustrate the concepts of angle bisectors and perpendicular bisectors, demonstrating how these segments relate to the triangle's geometry.

💡Vertex Angle

A vertex angle is the angle formed by two edges of a polygon that meet at a vertex. In the context of the video, the angle bisector bisects the vertex angle of a triangle, such as angle B in triangle ABC.

💡Incenter

The incenter is the point of concurrency where the angle bisectors of a triangle intersect. The video mentions that all three angle bisectors meet at the incenter, which is a key concept in understanding the symmetry and internal structure of triangles.

💡Midpoint

The midpoint of a line segment is the point that divides the segment into two equal parts. The script refers to the midpoint when discussing the perpendicular bisector, which passes through the midpoint of a side of the triangle.

💡Congruence

Congruence in geometry refers to the property of two figures being identical in shape and size. The video script uses the concept of congruence to explain that the segments created by an angle bisector or a perpendicular bisector are equal in length.

💡Circumcenter

The circumcenter is the point where the perpendicular bisectors of a triangle's sides intersect. It is mentioned in the video as the point of concurrency for the perpendicular bisectors, and it can be inside or outside the triangle.

💡Altitude

An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. The video script compares the perpendicular bisector to an altitude, noting that both form a 90-degree angle.

💡Median

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. The script mentions the median in the context of the incenter, noting that the median, like the angle bisector, intersects at a specific point within the triangle.

💡Congruency Postulate

The congruency postulate is a principle in geometry that states if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. The video script applies a similar concept when discussing the perpendicular bisector, stating that the segments on either side of the bisector are congruent.

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