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State Board · Class 8 · Mathematics

Lesson 4: Altitudes and Medians of a triangle

5 concepts · 40 practice questions with answers · Free to read

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Altitude of a triangle

From daily life

A mason holds a plumb-line from the top of a triangular roof gable straight down to the floor. The straight drop from the peak to the base is an altitude of the triangle.

Simple definition

The perpendicular segment from a vertex of a triangle to the opposite side is its altitude (शिरोलंब). Every triangle has three altitudes.

Rule 1: Recall and drawing altitudes — Recall: angle bisectors meet at the incentre I, and perpendicular bisectors of the sides meet at the circumcentre S. In ΔABC, seg AP ⊥ BC, so AP is the altitude on base BC. To draw the altitudes of ΔXYZ, use a set-square to drop XR ⊥ YZ, then YQ ⊥ XZ and ZP ⊥ XY.

IincentreScircumcentre

In the figure : In the figure the angle bisectors meet at the incentre I, and the perpendicular bisectors of the sides meet at the circumcentre S.

PABC

In the figure : In the figure altitude: the perpendicular from a vertex to the opposite side. Here seg AP ⊥ BC.

RQPXYZ

In the figure : In the figure the altitudes of ΔXYZ: XR ⊥ YZ, YQ ⊥ XZ and ZP ⊥ XY.

  • Easy : Easy: A triangle has 3 vertices, so 3 altitudes.
  • Medium : Medium: If the base is 10 cm and the area is 40 cm2, the altitude on it is 2 × 4010 = 8 cm.
  • Exam-type : Exam: In an isosceles triangle the altitude from the apex also bisects the base.
Remember

An altitude makes 90° with the opposite side.

Common mistake

Joining a vertex to the midpoint and calling it an altitude. That is a median.

Shortcut

Memory clue: "शिरोलंब = लंब, 90°".

Spot the difference

SegmentProperty
Altitude⊥ opposite side
Number in a triangle3
ToolSet-square

Practice questions

Easy
  1. 1What is an altitude?
  2. 2How many altitudes does a triangle have?
  3. 3What angle does an altitude make with the base?
Medium
  1. 4Which tool draws an altitude?
  2. 5Base 12 cm, area 30 cm2: altitude?
  3. 6How is an altitude different from a median?
Challenging
  1. 7In ΔABC, AP ⊥ BC. Which is the base?
  2. 8In which triangle do all the altitudes equal the medians?
Show answers (try them yourself first)
  1. 1The perpendicular from a vertex to the opposite side.
  2. 2Three.
  3. 390°.
  4. 4A set-square.
  5. 52 × 3012 = 5 cm.
  6. 6Altitude goes at 90°; median goes to the midpoint.
  7. 7BC.
  8. 8In an equilateral triangle.

Orthocentre

From daily life

Stand three poles at the corners of a triangle and tie a rope from each pole straight to the opposite wall at 90°. All three ropes cross at one single knot.

Simple definition

The three altitudes are concurrent (संगामी); the point where they meet is the orthocentre (लंबकेंद्र), denoted O.

Rule 1: Finding O — In ΔXYZ the altitudes XR, YQ and ZP pass through one point. That point is O. To find O, draw any two altitudes; the third passes through the same point (use it as a check).

OABC

In the figure : In the figure the three altitudes pass through one point, the orthocentre O.

  • Easy : Easy: Two altitudes are enough to locate O.
  • Medium : Medium: The third altitude is a check: it must pass through O.
  • Exam-type : Exam: Concurrent means passing through one common point.
Remember

Altitudes meet at the orthocentre O.

Common mistake

Confusing O with the centroid G. O is for altitudes, G is for medians.

Shortcut

Memory clue: "O = Orthocentre = शिरोलंब".

Spot the difference

PointMade by
Orthocentre OAltitudes
Centroid GMedians
Incentre IAngle bisectors

Practice questions

Easy
  1. 1What does concurrent mean?
  2. 2What is the orthocentre?
  3. 3Which letter denotes the orthocentre?
Medium
  1. 4How many altitudes do we need to find O?
  2. 5Which point comes from the angle bisectors?
  3. 6Which point comes from the perpendicular bisectors of the sides?
Challenging
  1. 7Draw an acute triangle: where is O?
  2. 8Why is the third altitude a good check?
Show answers (try them yourself first)
  1. 1Passing through one common point.
  2. 2The meeting point of the three altitudes.
  3. 3O.
  4. 4Two.
  5. 5The incentre.
  6. 6The circumcentre.
  7. 7Inside the triangle.
  8. 8Because all three must meet at one point.

Location of the orthocentre

From daily life

A tall tent pole sits in the middle of a tent, but for a very flat tent the poles lean outside. In the same way, where the altitudes meet depends on the shape of the triangle.

Simple definition

The position of the orthocentre O depends on the type of triangle: right → at the right-angle vertex, acute → inside, obtuse → outside.

Rule 1: Three cases — Right angled (sides 3, 4, 5): O is the vertex of the right angle. Acute angled (angles 60°, 70°, 50°): O is inside. Obtuse angled (one angle 110°): O is outside; the altitudes themselves do not meet, but the lines containing them (extended, dashed) meet at O.

Oright angledOobtuse angledOacute angled

In the figure : In the figure right angled: O is the vertex of the right angle. Obtuse angled: O is outside the triangle. Acute angled: O is inside the triangle.

DEFOABC

In the figure : In the figure in an obtuse angled triangle the altitudes themselves do not meet, but the lines containing them (dashed) meet at O outside the triangle.

  • Easy : Easy: Angles 40°, 60°, 80° are all acute, so O is inside.
  • Medium : Medium: Angles 25°, 35°, 120°: one obtuse angle, so O is outside.
  • Exam-type : Exam: Sides 5, 12, 13: 25 + 144 = 169, so it is right angled and O is at the vertex between 5 and 12.
Remember

Right: vertex. Acute: inside. Obtuse: outside (extend the altitudes).

Common mistake

Looking for O inside an obtuse triangle. Extend the altitudes with dashed lines instead.

Shortcut

Memory clue: "काटकोन-कोपरा, लघु-आत, विशाल-बाहेर".

Spot the difference

TrianglePosition of O
Right angledAt the right-angle vertex
Acute angledInside
Obtuse angledOutside

Practice questions

Easy
  1. 1Where is O in a right angled triangle?
  2. 2Where is O in an acute angled triangle?
  3. 3Where is O in an obtuse angled triangle?
Medium
  1. 4Angles 50°, 60°, 70°: where is O?
  2. 5Angles 30°, 40°, 110°: where is O?
  3. 6Sides 6, 8, 10: where is O?
Challenging
  1. 7Why do we extend the altitudes of an obtuse triangle?
  2. 8In a right triangle, two altitudes are the two legs. Why?
Show answers (try them yourself first)
  1. 1At the vertex of the right angle.
  2. 2Inside.
  3. 3Outside.
  4. 4Inside.
  5. 5Outside.
  6. 6Right angled (36 + 64 = 100), so at the vertex between 6 and 8.
  7. 7The altitudes themselves do not meet; their lines do, outside.
  8. 8Each leg is perpendicular to the other, so it is the altitude on it.

Median and centroid

From daily life

A school cook wants to cut a triangular flatbread so that each piece is exactly balanced. She joins each corner to the middle of the opposite side. These lines are medians.

Simple definition

A median (मध्यगा) joins a vertex to the midpoint of the opposite side. The three medians meet at the centroid (केंद्रक) G, always inside the triangle.

Rule 1: Drawing medians — In ΔHCF, D is the midpoint of CH, so FD is a median. To draw the medians of ΔABC, mark midpoints P, Q, R of AB, BC, AC and draw CP, AQ, BR. They are concurrent at G.

DFCH

In the figure : In the figure median: the segment from a vertex to the midpoint of the opposite side. D is the midpoint of CH (equal ticks), so FD is a median.

QRPGABC

In the figure : In the figure the medians AQ, BR and CP (P, Q, R are the midpoints of AB, BC, AC) are concurrent at the centroid G.

Gright angledGobtuse angledGacute angled

In the figure : In the figure the centroid G is in the interior of every triangle: right angled, obtuse angled and acute angled.

Gpin at G

In the figure : In the figure a cardboard triangle balances on a pin placed at its centroid G.

  • Easy : Easy: A triangle has 3 medians.
  • Medium : Medium: The midpoint of a 14 cm side divides it into 7 cm and 7 cm.
  • Exam-type : Exam: The centroid is always inside, unlike the orthocentre which can be outside.
Remember

Median → midpoint. Centroid G is the balance point and is always inside.

Common mistake

Calling the altitude a median. The median goes to the midpoint, not at 90°.

Shortcut

Memory clue: "मध्यगा = मध्यबिंदू, G = तोल".

Spot the difference

SegmentGoes to
AltitudeOpposite side at 90°
MedianMidpoint of opposite side
Centroid GBalance point inside

Practice questions

Easy
  1. 1What is a median?
  2. 2What is the centroid?
  3. 3Which letter denotes the centroid?
Medium
  1. 4Can the centroid be outside a triangle?
  2. 5If D is the midpoint of CH = 18 cm, find CD.
  3. 6Why does a cardboard triangle balance at G?
Challenging
  1. 7In an equilateral triangle, is each altitude also a median?
  2. 8How many medians do we draw to find G?
Show answers (try them yourself first)
  1. 1A segment from a vertex to the midpoint of the opposite side.
  2. 2The meeting point of the three medians.
  3. 3G.
  4. 4No, always inside.
  5. 59 cm.
  6. 6G is its balance (centre of mass) point.
  7. 7Yes.
  8. 8Two are enough.

Property of the centroid

From daily life

If you balance a ruler on a finger, the ruler tilts if the finger is not at the centre. A triangle balances at its centroid, and the centroid splits each median into a 2-part piece and a 1-part piece.

Simple definition

The centroid divides each median in the ratio 2 : 1, the longer part towards the vertex: AG : GR = 2 : 1.

Rule 1: Using 2 : 1 — Vertex part = 23 of the median and side part = 13. If AG = 12 then GR = 6. If GP = 5 then CG = 10. If BG = 8 then BQ = 12 (8 + 4). If the whole median is 24, then AG = 16 and GR = 8.

RG147ABC

In the figure : In the figure the centroid divides each median in the ratio 2 : 1. If l(AG) = 14 then l(GR) = 7: the part from the vertex is twice the part from G to the side.

QRPG212121ABC

In the figure : In the figure on every median the part from the vertex to G is 2 units and the part from G to the side is 1 unit.

RG30 cm20 cm10 cmABC

In the figure : In the figure if the whole median l(AR) = 30 cm, then l(AG) = 23 × 30 = 20 cm and l(GR) = 13 × 30 = 10 cm.

  • Easy : Easy: GR = 3 means AG = 6.
  • Medium : Medium: Whole median 27 cm: AG = 18 cm and GR = 9 cm.
  • Exam-type : Exam: AG = 14 cm gives GR = 7 cm and the whole median AR = 21 cm.
Remember

Vertex side : base side = 2 : 1. Total parts = 3.

Common mistake

Putting 2 on the part near the side. The longer part is near the vertex.

Shortcut

Memory clue: "शिरोबिंदूकडे 2, बाजूकडे 1".

Spot the difference

GivenFindAnswer
GR = 5AG10
AG = 12Whole median18
Whole median 30AG, GR20, 10

Practice questions

Easy
  1. 1In what ratio does G divide a median?
  2. 2Which part is longer?
  3. 3GR = 4. Find AG.
Medium
  1. 4AG = 10. Find GR.
  2. 5Whole median = 45 cm. Find AG.
  3. 6BG = 6. Find the whole median BQ.
Challenging
  1. 7AP = 6 and G is the centroid on AP. Find AG and GP.
  2. 8Fraction of a median from the vertex to G?
Show answers (try them yourself first)
  1. 12 : 1.
  2. 2The one towards the vertex.
  3. 38.
  4. 45.
  5. 530 cm.
  6. 69.
  7. 7AG = 4, GP = 2.
  8. 823.

Quick revision

  • Altitude: perpendicular from a vertex to the opposite side; the altitudes meet at the orthocentre O.
  • Orthocentre: right angled triangle → the vertex of the right angle; obtuse angled → exterior; acute angled → interior.
  • Median: vertex to the midpoint of the opposite side; the medians meet at the centroid G, which divides each median in the ratio 2 : 1.
Live figure

Quick revision

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