# SPECIAL LINE SEGMENTS OF TRIANGLES

Special Line Segments of Triangles :

The following are the four special line segments of a triangle.

1. Angle bisector

2. Perpendicular bisector

3. Median

4. Altitude

## Special Line Segments of Triangles

Angle Bisector :

The angle bisector divides an angle into two equal parts.

In the above triangle ABC, the line AD bisects the angle A.

So, AD is the angle bisector of angle A.

Similarly we can draw angle bisectors to the angles B and C.

Hence, every triangle will have three angle bisectors.

Perpendicular Bisector :

The perpendicular bisector of a triangle is a line which is passing through the mid point of the side and also perpendicular to that side.

In the above triangle, the line ED is passing through the mid point of the side AB and also it is perpendicular to the side AB

So, ED is the perpendicular bisector in the above triangle.

Similarly we can draw perpendicular bisector to the sides BC and AC.

Hence, every triangle will have three perpendicular bisectors.

Median :

The median of a triangle is a line segment joining joining a vertex to the mid point of the opposite side.

In the above triangle, the line segment joining the vertex C and the mid point of AB which is D.

So, CD is the median in the above triangle.

Similarly, we can draw medians from the vertices A and B also.

Hence, every triangle will have three medians.

Altitude (Height) :

Altitude of a triangle is a line segment through a vertex and perpendicular to (That is, forming a right angle with) the opposite side which is considered to be base.

In the above triangle, the line segment AD  is passing through the vertex A and perpendicular to the side BC.

So, AD is the altitude of the triangle.

Similarly, we can draw altitudes from the vertices B and C also.

Hence, every triangle will have three altitudes.

## Special Line Segments of Triangles - Examples

Example 1 :

In the triangle ABC, angle A is divided in to equal halves by the line segment AC. What type of line segment is AC ?

Solution :

Given :  Angle A is divided in to equal halves by the line segment AC.

Only the angle bisector will divide an angle in to equal halves.

So, the line segment AC is angle bisector.

Example 2 :

In the triangle ABC, the line segment AD is passing through the vertex A and also perpendicular to the opposite side of the vertex A. What type of line segment is AD ?

Solution :

Given : The line segment AD is passing through the vertex A and also perpendicular to the opposite side of the vertex A.

Only the altitude will pass through the vertex and perpendicular to the opposite side of the vertex.

So, the line segment AD is altitude.

Example 3 :

In the triangle ABC, the line segment ED is not passing through any of the vertex but it is perpendicular to one of the sides of the triangle and also passing through the mid point of the same side. What type of line segment is ED ?

Solution :

Given : The line segment ED is not passing through any of the vertex but it is perpendicular to one of the sides of the triangle and also passing through the mid point of the same side

Since the line segment ED is not passing through any of the vertex, it can not be altitude.

But ED is perpendicular to one of the sides of the triangle and also it is passing through the midpoint of the same side.

So, the line segment ED is perpendicular bisector.

Example 4 :

In the triangle ABC, the line segment AD is passing through the vertex A, perpendicular to the side BC and also passing through the mid point of the side BC. What type of line segment is AD ?

Solution :

Given : The line segment AD is passing through the vertex A, perpendicular to the side BC and also passing through the mid point of the side BC

Since the line segment AD is passing through one of the vertices vertex and perpendicular to the side BC, clearly AD is altitude.

But AD is also passing through the midpoint of BC. So it is also perpendicular bisector and median.

So, the line segment AD is altitude,  perpendicular bisector and median.

Example 5 :

In the triangle ABC, the line segment AD joining the vertex A and the midpoint of the side BC.  What type of line segment is AD ?

Solution :

Given : The line segment AD joining the vertex A and the midpoint of the side BC.

Only the median will join one of the vertices of the triangle and the mid point of the opposite side of the vertex.

So, the line segment AD is median.

Example 6 :

In the triangle ABC, there is a angle bisector at angle A. After the angle A is divided in to two equal halves, if each half measures 22°, find angle A.

Solution :

Given : Angle A is divided in to two equal halves by the angle bisector and each half measures 22°,

Angle A  =  22° + 22°

Angle A  =  44°

Example 7 :

In the triangle given below, AD is the angle bisector. If B  =  70° and C  =  60°, find BAD and DAC.

Solution :

The angle sum property of a triangle states that the angles of a triangle always add up to 180°.

Then,

A + B + C  =  180°

A + 70° + 60°  =  180°

A + 130°  =  180°

A  =  50°

Because AD is the angle bisector, it divides the angle A into two equal halves. That is 25° and 25°

So,

DAC  =  25°

Example 8 :

In the triangle given below, ED is the perpendicular bisector. If B  =  60° and C  =  70°, find angle AED.

Solution :

The angle sum property of a triangle states that the angles of a triangle always add up to 180°.

A + B + C  =  180°

A + 60 + 70  =  180

A + 130  =  180

A  =  50°

In triangle AED, we have

A + AED + EDA  =  180°

50 + AED + 90  =  180

AED + 140  =  180

AED  =  40°

Example 9 :

In the triangle given below, CD is the median. If B  = 60° and ADC  =  120°, find BCD.

Solution :

From the given figure, angle ADC and angle CDB together form a straight angle.

Then,

120° + CDB  =  180°

CDB  =  60°

In triangle BCD, we have

B + BCD + CDB  =  180°

60° + BCD + 60°  =  180°

BCD + 120°  =  180°

BCD  =  60°

Example 10 :

In the triangle given below, AD  =  4 cm and BC  =  8 cm, find the area of the triangle.

Solution :

In the given figure, clearly AD is the height (altitude) and BC is the base of the triangle.

Area of the triangle  =  (1/2)  b  h

=  (1/2)  8  4

=  16 square cm.

So, the area of the triangle is 16 square cm.

After having gone through the stuff given above, we hope that the students would have understood special line segments in triangles.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

You can also visit our following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6
The text passage provides an overview of the four special line segments of a triangle (angle bisector, perpendicular bisector, median, and altitude) and includes several examples that involve identifying these line segments and solving related problems.
multiple_questions_with_answers

- example 1: identifies the type of line segment (angle bisector).
- example 2: identifies the type of line segment (altitude).
- example 3: identifies the type of line segment (perpendicular bisector).
- example 4: identifies the type of line segment (altitude, perpendicular bisector, and median).
- example 5: identifies the type of line segment (median).
- example 6: solves for the measure of angle a (44°).
- example 7: solves for the measures of angles bad and dac (25° each).
- example 8: solves for the measure of angle aed (40°).
- example 9: solves for the measure of angle bcd (60°).
- example 10: solves for the area of the triangle (16 square cm).
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# SPECIAL LINE SEGMENTS OF TRIANGLES

Special Line Segments of Triangles :

The following are the four special line segments of a triangle.

1. Angle bisector

2. Perpendicular bisector

3. Median

4. Altitude

## Special Line Segments of Triangles

Angle Bisector :

The angle bisector divides an angle into two equal parts.

In the above triangle ABC, the line AD bisects the angle A.

So, AD is the angle bisector of angle A.

Similarly we can draw angle bisectors to the angles B and C.

Hence, every triangle will have three angle bisectors.

Perpendicular Bisector :

The perpendicular bisector of a triangle is a line which is passing through the mid point of the side and also perpendicular to that side.

In the above triangle, the line ED is passing through the mid point of the side AB and also it is perpendicular to the side AB

So, ED is the perpendicular bisector in the above triangle.

Similarly we can draw perpendicular bisector to the sides BC and AC.

Hence, every triangle will have three perpendicular bisectors.

Median :

The median of a triangle is a line segment joining joining a vertex to the mid point of the opposite side.

In the above triangle, the line segment joining the vertex C and the mid point of AB which is D.

So, CD is the median in the above triangle.

Similarly, we can draw medians from the vertices A and B also.

Hence, every triangle will have three medians.

Altitude (Height) :

Altitude of a triangle is a line segment through a vertex and perpendicular to (That is, forming a right angle with) the opposite side which is considered to be base.

In the above triangle, the line segment AD  is passing through the vertex A and perpendicular to the side BC.

So, AD is the altitude of the triangle.

Similarly, we can draw altitudes from the vertices B and C also.

Hence, every triangle will have three altitudes.

## Special Line Segments of Triangles - Examples

Example 1 :

In the triangle ABC, angle A is divided in to equal halves by the line segment AC. What type of line segment is AC ?

Solution :

Given :  Angle A is divided in to equal halves by the line segment AC.

Only the angle bisector will divide an angle in to equal halves.

So, the line segment AC is angle bisector.

Example 2 :

In the triangle ABC, the line segment AD is passing through the vertex A and also perpendicular to the opposite side of the vertex A. What type of line segment is AD ?

Solution :

Given : The line segment AD is passing through the vertex A and also perpendicular to the opposite side of the vertex A.

Only the altitude will pass through the vertex and perpendicular to the opposite side of the vertex.

So, the line segment AD is altitude.

Example 3 :

In the triangle ABC, the line segment ED is not passing through any of the vertex but it is perpendicular to one of the sides of the triangle and also passing through the mid point of the same side. What type of line segment is ED ?

Solution :

Given : The line segment ED is not passing through any of the vertex but it is perpendicular to one of the sides of the triangle and also passing through the mid point of the same side

Since the line segment ED is not passing through any of the vertex, it can not be altitude.

But ED is perpendicular to one of the sides of the triangle and also it is passing through the midpoint of the same side.

So, the line segment ED is perpendicular bisector.

Example 4 :

In the triangle ABC, the line segment AD is passing through the vertex A, perpendicular to the side BC and also passing through the mid point of the side BC. What type of line segment is AD ?

Solution :

Given : The line segment AD is passing through the vertex A, perpendicular to the side BC and also passing through the mid point of the side BC

Since the line segment AD is passing through one of the vertices vertex and perpendicular to the side BC, clearly AD is altitude.

But AD is also passing through the midpoint of BC. So it is also perpendicular bisector and median.

So, the line segment AD is altitude,  perpendicular bisector and median.

Example 5 :

In the triangle ABC, the line segment AD joining the vertex A and the midpoint of the side BC.  What type of line segment is AD ?

Solution :

Given : The line segment AD joining the vertex A and the midpoint of the side BC.

Only the median will join one of the vertices of the triangle and the mid point of the opposite side of the vertex.

So, the line segment AD is median.

Example 6 :

In the triangle ABC, there is a angle bisector at angle A. After the angle A is divided in to two equal halves, if each half measures 22°, find angle A.

Solution :

Given : Angle A is divided in to two equal halves by the angle bisector and each half measures 22°,

Angle A  =  22° + 22°

Angle A  =  44°

Example 7 :

In the triangle given below, AD is the angle bisector. If B  =  70° and C  =  60°, find BAD and DAC.

Solution :

The angle sum property of a triangle states that the angles of a triangle always add up to 180°.

Then,

A + B + C  =  180°

A + 70° + 60°  =  180°

A + 130°  =  180°

A  =  50°

Because AD is the angle bisector, it divides the angle A into two equal halves. That is 25° and 25°

So,

DAC  =  25°

Example 8 :

In the triangle given below, ED is the perpendicular bisector. If B  =  60° and C  =  70°, find angle AED.

Solution :

The angle sum property of a triangle states that the angles of a triangle always add up to 180°.

A + B + C  =  180°

A + 60 + 70  =  180

A + 130  =  180

A  =  50°

In triangle AED, we have

A + AED + EDA  =  180°

50 + AED + 90  =  180

AED + 140  =  180

AED  =  40°

Example 9 :

In the triangle given below, CD is the median. If B  = 60° and ADC  =  120°, find BCD.

Solution :

From the given figure, angle ADC and angle CDB together form a straight angle.

Then,

120° + CDB  =  180°

CDB  =  60°

In triangle BCD, we have

B + BCD + CDB  =  180°

60° + BCD + 60°  =  180°

BCD + 120°  =  180°

BCD  =  60°

Example 10 :

In the triangle given below, AD  =  4 cm and BC  =  8 cm, find the area of the triangle.

Solution :

In the given figure, clearly AD is the height (altitude) and BC is the base of the triangle.

Area of the triangle  =  (1/2)  b  h

=  (1/2)  8  4

=  16 square cm.

So, the area of the triangle is 16 square cm.

After having gone through the stuff given above, we hope that the students would have understood special line segments in triangles.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

You can also visit our following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6
user
The text passage provides an overview of the four special line segments of a triangle (angle bisector, perpendicular bisector, median, and altitude) and includes several examples that involve identifying these line segments and solving related problems.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS

- Example 1: Identifies the type of line segment (angle bisector).
- Example 2: Identifies the type of line segment (altitude).
- Example 3: Identifies the type of line segment (perpendicular bisector).
- Example 4: Identifies the type of line segment (altitude, perpendicular bisector, and median).
- Example 5: Identifies the type of line segment (median).
- Example 6: Solves for the measure of angle A (44°).
- Example 7: Solves for the measures of angles BAD and DAC (25° each).
- Example 8: Solves for the measure of angle AED (40°).
- Example 9: Solves for the measure of angle BCD (60°).
- Example 10: Solves for the area of the triangle (16 square cm).