9394949438

[LATEST]$type=sticky$show=home$rm=0$va=0$count=4$va=0

PROPERTIES OF TRIANGLES, HEIGHTS AND DISTANCES

REPRESENTATIONS IN A TRIANGLE

In any triangle ABC, the side BC, opposite to the angle A is denoted by a, the side CA and AB opposite to the angles B and C respectively are denoted by b and c respectively.
Semi perimeter of the triangles is denoted by s and its area by Δ or S. In this unit, we shall discuss various relations between the sides a, b, c, and the angles A, B, C, of ΔABC.
Further, in geometry we are familiar with the following relations
  • A + B + C = π
  • a + b > c
  • b + c > a
  • c + a > b

SINE RULE

In any Δ ABC, the sines  of the angles are proportional to the lengths of the opposite sides,
i.e.

Note 1 : The above rule can also be written as

Note 2 : The rule is generally used to express sides of the triangle in terms of sines of angles and vice-versa as discussed below.
Let
Then a = k sin A, b = k sin B, c = k sin C or
.

COSINE FORMULAE

In any Δ ABC,

PROJECTION FORMULAE

In any Δ ABC,
a = b cos C + c cos B
b = c cos A + a cos C
c = a cos B + b cos A

TRIGONOMETRIC RATIOS OF HALF OF THE ANGLES OF A TRIANGLE

In any Δ ABC, we have
  •         

AREA OF A TRIANGLE

,
Similarly,and

NAPIER’S ANALOGY

In any triangle ABC
,

EXPRESSING SINES OF ANGLES IN TERMS OF SIDES AND AREA

EXPRESSING TANGENTS AND COTANGENTS OF ANGLES IN TERMS OF SIDES AND AREA

  • ,
  • ,
  • ,

COTANGENT RULES

On the base BC of the triangle ABC, take a point D, such that
BD : DC = m : n, ∠BAD = α , ∠CAD = β and ∠ADB = θ, then
(m + n) cot θ = n cot β – m cot α
(m + n) cot θ = m cot C – n cot B
These results are frequently used in statics.

CIRCLES CONNECTED TO A TRIANGLE

CIRCUMCIRCLE OF A TRIANGLE

The circle, which passes through the angular points of a triangle ABC, is called its circumcircle. The centre of this circle is the point of intersection of perpendicular bisectors of the sides and is called the circumcentre. Its radius is always denoted by R. The circumcentre may lie within, outside or upon one of the sides of the triangle.

Circumradius : The radius of the circumcircle of a triangle ABC is called the circumradius given by

INSCRIBED CIRCLE OR INCIRCLE OF A TRIANGLE

The circle, which can be inscribed within the triangle so as to touch each of its sides, is called its inscribed circle or incircle. The centre of this circle is the point of intersection of bisectors of the angles of the triangle. The radius of the circle is always denoted by r and is equal to the length of the perpendicular from its centre to any of the sides of triangle.

Inradius : The radius of the inscribed circles of a triangle is called the in-radius. It is denoted by r and is given by

ESCRIBED CIRCLES OF A TRIANGLE

The circle, which touches the side BC from outside and two sides AB and AC produced of a triangle ABC is called the escribed circle opposite to the angle A. Its radius is denoted by r1. Similarly r2 and r3 denote the radii of the escribed circles opposite to the angles B and C respectively.

The centres of the escribed circles are called the ex–centres. The centre of the escribed circle opposite to the angle A is the point of intersection of the external bisectors of angles B and C. The internal bisector of A also passes through the same point. This centre is generally denoted by I1.

FORMULAE FOR r1, r2, r3
In any , we have
  • r1 = , r2= , r3 =
  • r1 = ;  r2= ;  r3 =
  • r1 = ;   r2= ;   r3 =
  • r1 = 4 R sincoscos;  r2= 4 R cossincos;  r3 = 4 R coscos sin

ORTHOCENTRE OF A TRIANGLE

Let the perpendiculars AL, BM and CN from the vertices A, B and C on the opposite sides BC, CA and AB of ΔABC, respectively, meet at O′. Then O′ is the orthocentre of the ΔABC
The triangle LMN is called the pedal triangle of the  ΔABC.

IMPORTANT RESULTS
  1. The distances of the orthocentre of the triangle from the vertices are 2RcosA, 2RcosB, 2RcosC and its distances from the sides are 2RcosB cosC, 2RcosC cosA, 2RcosA cosB.
  2. Circumradius of the pedal triangle
  3. Area of the pedal triangle
  4. Lengths of the medians AD, BE and CF of ΔABC are given by
  1. Circumcentre O, centroid G and orthocentre O' are collinear and G divides OO′ in the ratio 1 : 2.
  2. Distance between the circumcentre O and the incentre I is
Where I1, I2, I3 are the centres of the escribed circles opposite to the angles A, B and C respectively.

CYCLIC QUADRILATERAL

A quadrilateral is a cyclic quadrilateral if its vertices lie on a circle.
  1. Area of cyclic quadrilateral ABCD is
where
  1. Circumradius of cyclic quadrilateral
  1. and similarly other angles.
  2. Ptolemy’s Theorem : If ABCD is a cyclic quadrilateral, then AC.BD = AB.CD + BC.AD
i.e in a cyclic quadrilateral the sum of the products of the lengths of opposite sides is equal to the product of its diagonals.

REGULAR POLYGON

A polygon is called a regular polygon if all its sides are equal and its angles are equal.

Note :
  1. If a polygon has ‘n’ sides, sum of its internal angles is and each angle is .
  2. In a regular polygon the centroid, the circumcentre and the incentre are same.
  3. Area of a regular polygon where a is length of side, n is number of sides of polygon, R is radius of circumscribing circle and r is the radius of incircle of the polygon.

SOLUTION OF TRIANGLES

In a triangle there are six elements, viz., three sides and three angles. In plane geometry we have done that if three of the elements are given at least one of which must be a side, then the other three elements can be uniquely determined. The procedure of determining unknown elements from the known elements is called solving a triangle.

SOLUTION OF A RIGHT ANGLED TRIANGLE

Case I. When two sides are given:
Let the triangle be right angled at C. Then we can determine the remaining elements as given in the following table:
Given Required
(i) a, b tan A = , B = 900–A, c =
(ii) a, c Sin A = , B = 900–A, b =  c cos A

Case II. When a side and an acute angle are given:
In this case we can determine the remaining elements as given in the following table:
Given Required
(i) a, A B = 900–A, b = a cot A,  c =
(ii) c, A B = 900–A, a = c sin A, b =  c cos A

SOLUTION OF TRIANGLES IN GENERAL

Case I. When three sides  a, b and c are given:
In this case the remaining elements are determined by using the following formulae:
, where 2 s = a + b + c
sin A = , sin B = , sin C =
or tan = ,  tan = , tan =
And  A + B + C = 1800
Cosine formulae can also be used to find the angles.

Case II. When two sides a, b and the included angle C are given :
In this case we use the following formulae:
= a b sin C, tan = cot if a > b
tan = cot , if b > a  
= 900and

Case III. When one side a and two angles A and B are given:
In this case we use the following formulae to determine the remaining elements:
A + B + C = 1800  C = 1800–A – B
b = and c = ,   = a b sin C.

Case IV. When two sides  a, b and the angle A opposite to one side is given:
In this case we use the following formulae:
sin B = ...(I)
C = 1800– A – B,  c =
From (I), the following possibilities will arise:
  • When A is an acute angle and a < b sin A
In this case the relation gives that  sin B > 1, which is impossible. Hence no triangle is possible.
  • When A is an acute angle and a = b sin A. In this case only one triangle is possible which is right angled at B.
  • When A is an acute and a > b sin A. In this case there are two values of B given by , say, B1 and B2 such that .
Side c can be obtained by using . Hence two triangles are possible. It is called ambiguous case

SOME IMPORTANT RESULTS
  1. In any right-angled triangle, the orthocentre coincides with the vertex containing the right-angled.
  2. The mid-point of the hypotenuse of  a right-angled triangle is equidistant from the three vertices of the triangle.
  3. The midpoint of the hypotenuse of a right angled triangle is the circumcentre of the triangle.

HEIGHT AND DISTANCES

Let an observer at the point O is observing an object at the point P. The line OP is called the LINE OF SIGHT of the point P. Let OA be the horizontal line passing through the position of observer in the same vertical plane with OP. The acute angle AOP, between the line of sight and the horizontal line is known as the ANGLE OF ELEVATION of P, if the object P is above the horizontal plane and the ANGLE OF DEPRESSION of P, if P lies below the horizontal plane, as seen from O.
Obviously, the angle of elevation of P as seen from a point O is equal to the angle of depression of O as seen from P.

TO FIND THE HEIGHT AND THE DISTANCE OF AN INACCESSIBLE TOWER STANDING ON A HORIZONTAL PLANE

Let AB be the tower and B its foot. On the horizontal line through B, take two points P and Q so that the line PQ passes through B. Measure the length PQ.
Let PQ = a.
Let the angles of elevation of the top A of the tower as seen from P and Q be respectively α and β ( β > α) then
Let AB = x, BQ = y.
From Δ APB,


From Δ AQB,  
∴ y = x cot β
.
In the above case, P, Q are on the same side of the tower. If the two points are on the opposite sides of the tower then from the adjoining figure, we get
and
β need not be greater than α in this case.
Note Here, all the lines AP, AQ, AB are in one plane.

TO FIND THE DISTANCE BETWEEN TWO INACCESSIBLE OBJECTS OR POINTS

Let the distance between two points P and Q be required. Let A and B be two convenient points. Now measure the length AB and calculate the angles PAB, QAB, PBA and QBA.
Let
If the lines AP, AQ, AB are coplanar then
If however the lines are not coplanar then measure the ∠PAQ which may not be equal to α – β In any case, we may consider the ∠PAQ as known.
Now, from ΔPAB,
From ΔAQB,
Thus, the sides AP, AQ and the included angle PAQ of the Δ PAQ are determined. Hence AQ can be obtained by the formula :

BEARING OF A LINE

The bearing of a horizontal line (i.e., a line in a horizontal plane) is the positive acute angle made by this line with the north-south line in the same horizontal plane.
The amount of eastward or westward deviation of a line from the north-south line measures the bearing of this line. Thus the bearing of the north-south line is zero and that of the east-west line is 90°. In giving the bearing of a line  the letter. N or S is written before the angle to show if the line is measured from the northern part or from the southern part of the north-south line and the letter E or W follows the value of the angle to indicate if the line lies to the east or to the west of the north-south line.

Note :
  • In the above figure the bearings of the lines OA, OB, OC and OD are respectively N 40°E, S 35°E, S 75° W and N 15°W. Sometimes N 40ºE means the bearing of the point A from the point O. Likewise, for the other points B, C, D, etc.
  • Sometimes the acute angle between two lines is called the bearing of one line with respect to the other.

HEIGHTS AND DISTANCES IN THREE DIMENSIONS

PERPENDICULARITY OF A LINE AND A PLANE
  • If a line l is perpendicular to a plane π then the line l is perpendicular to any line m in the plane π [see fig. (i)].
[Fig. (i)]                          [fig. (ii)]
  • If a line l is perpendicular to two intersecting lines m and n then the line l is perpendicular to the plane π passing through the lines m and n [see Fig. (ii)]
  • PQ is perpendicular to the horizontal plane and the lines QA, QB, QC, .......are in the plane. Then PQ丄QA, PQ丄QB, PQ丄QC,..... Hence the triangles PQA, PQB, PQC..... are all right angled triangles where ∠PQA = 90°, ∠PQB = 90°, ∠PQC = 90°,…….
If the angles of  elevation of P at A, B, C, .... are α, β, γ ... then QA = h cot α, QB = h cot β, QC = hcotγ,....

TO FIND THE HEIGHT AND THE DISTANCE OF AN INACCESSIBLE TOWER STANDING ON A HORIZONTAL PLANE.
i.e., measurement in more than one plane
Let AB be a tower and P be any point on the horizontal plane passing through the foot of the tower.
Measure the distance PQ in any suitable direction and also measure the angles ∠APB, ∠APQ, ∠AQP.
Let PQ = a,  ∠APB = α, ∠APQ = β, ∠AQP = γ
From
  
Again, from Δ APB,
If PB = y then cos α = y /AP

Want to know more

Want to Know More
Please fill in the details below:

INNER POST ADS

Latest IITJEE Articles$type=three$c=3$author=hide$comment=hide$rm=hide$date=hide$snippet=hide

Latest NEET Articles$type=three$c=3$author=hide$comment=hide$rm=hide$date=hide$snippet=hide

Name

Alternating Current,60,Basic Maths,2,best books for iit jee,2,best coaching institute for iit,1,best coaching institute for iit jee preparation,1,best iit jee coaching delhi,1,best iit jee coaching in delhi,2,best study material for iit jee,4,Blog,62,books for jee preparation,1,books recommended by iit toppers,3,Capacitance,3,CBSE NEET,9,cbse neet 2019,3,CBSE NEET 2020,1,cbse neet nic,1,Centre of Mass,2,Chemistry,58,Class 12 Physics,15,coaching for jee advanced,1,coaching institute for iit jee,2,Collision,2,Current Electricity,4,Electromagnetic Induction,3,Electronics,1,Electrostatics,3,Energy,1,Fluid Mechanics,4,Gravitation,2,Heat,4,iit admission,1,iit advanced,1,iit coaching centre,3,iit coaching centre in delhi,2,iit coaching classes,2,iit coaching in delhi,1,iit coaching institute in delhi,1,iit entrance exam,1,iit entrance exam syllabus,2,iit exam pattern,2,iit jee,4,iit jee 2019,3,iit jee advanced,2,iit jee books,3,iit jee coaching,2,iit jee exam,3,iit jee exam 2019,1,iit jee exam pattern,3,iit jee institute,1,iit jee main 2019,2,iit jee mains,2,iit jee mains syllabus,2,iit jee material,1,iit jee online test,3,iit jee practice test,3,iit jee preparation,6,iit jee preparation in delhi,2,iit jee preparation time,1,iit jee preparation tips by toppers,2,iit jee question paper,1,iit jee study material,3,iit jee study materials,2,iit jee syllabus,2,iit jee syllabus 2019,2,iit jee test,3,iit preparation,2,iit preparation books,5,iit preparation time table,2,iit preparation tips,2,iit syllabus,2,iit test series,3,IITJEE,100,JEE Advanced,82,jee advanced exam,1,jee advanced exam pattern,1,jee advanced paper,1,JEE Books,1,JEE Coaching Delhi,3,jee exam,3,jee exam 2019,6,JEE Exam Pattern,2,jee exam pattern 2019,1,jee exam preparation,1,JEE Main,85,jee main 2019,4,JEE Main application form,1,jee main coaching,1,JEE Main eligibility criteria,3,jee main exam,1,jee main exam 2019,3,jee main online question paper,1,jee main online test,3,jee main registration,2,jee main syllabus,2,jee mains question bank,1,jee mains test papers,3,JEE Mock Test,2,jee notes,1,jee past papers,1,JEE Preparation,2,jee preparation in delhi,1,jee preparation material,4,JEE Study Material,1,jee syllabus,6,JEE Syllabus Chemistry,1,JEE Syllabus Maths,1,JEE Syllabus Physics,1,jee test series,3,Kinematics,1,Latest article,1,Latest Articles,61,latest news about neet exam,1,Laws of Motion,2,Magnetic Effect of Current,3,Magnetism,3,Modern Physics,1,NCERT Solutions,15,neet,2,neet 2019,1,neet 2019 eligibility criteria,1,neet 2019 exam date,2,neet 2019 test series,2,NEET 2020 Eligibility Criteria,1,neet application form 2019 last date,1,Neet Biology Syllabus,1,Neet Books,3,neet eligibility criteria,3,neet exam 2019,7,neet exam application,1,neet exam date,1,neet exam details,1,neet exam pattern,6,neet exam pattern 2019,2,neet examination,1,neet mock test 2019,1,Neet Notes,3,Neet Online Application Form,3,neet online test,2,neet past papers,1,neet physics syllabus,1,neet practice test,2,NEET preparation books,1,neet qualification marks,1,NEET question paper 2019,1,neet question papers,1,neet registration,1,Neet Study Material,3,neet syllabus,6,neet syllabus 2019,5,NEET Syllabus 2020,1,neet syllabus chemistry,1,neet syllabus for biology,1,neet syllabus for physics,1,neet test series,1,neet ug 2019,2,news,1,online study material for iit jee,1,Optical Instruments,1,Physics,110,physics books for iit jee,1,Power,1,Practical Physics,1,Ray Optics,1,Rotational Motion,3,SHM,3,Simple Harmonic Motion,3,study materials for iit jee,1,Study Notes,110,study notes for iit jee,1,Thermodynamics,4,Units and Dimensions,1,Vectors,2,Wave Motion,3,Wave Optics,1,Work,1,
ltr
item
Best Coaching Institute in Delhi for IIT JEE, NEET, AIIMS, PMT - Clear IIT Medical: Properties of Triangles, Height & Distances | Mathematics Notes for IITJEE Main
Properties of Triangles, Height & Distances | Mathematics Notes for IITJEE Main
https://lh4.googleusercontent.com/6b1sZZnk0-Z7ukdZ9cb5c8iim3bUGs2TmPMwV9FEQlJYVqF_k06H8K3U_DvKDmWK4lfLEyfEuf-wTHIydwaKsgAc9-i-jZoSQjCjhVfyyFUQg-JXgjgURlJ13Rntk7jOjZNBqLU
https://lh4.googleusercontent.com/6b1sZZnk0-Z7ukdZ9cb5c8iim3bUGs2TmPMwV9FEQlJYVqF_k06H8K3U_DvKDmWK4lfLEyfEuf-wTHIydwaKsgAc9-i-jZoSQjCjhVfyyFUQg-JXgjgURlJ13Rntk7jOjZNBqLU=s72-c
Best Coaching Institute in Delhi for IIT JEE, NEET, AIIMS, PMT - Clear IIT Medical
https://www.cleariitmedical.com/2019/06/mathematics-notes-properties-of-triangles-and-height-and-distances.html
https://www.cleariitmedical.com/
https://www.cleariitmedical.com/
https://www.cleariitmedical.com/2019/06/mathematics-notes-properties-of-triangles-and-height-and-distances.html
true
7783647550433378923
UTF-8
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy