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State Board · Class 6 · Mathematics

Lesson 14: Geometrical Constructions

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

This lesson is also in: मराठी

Geometry tools

From daily life

Open your geometry box. Each tool does one job: one draws circles, one draws straight lines, one measures angles. Knowing the right tool for the job makes drawings neat and exact.

Simple definition

A compass (कंपास) draws circles and arcs, a ruler (पट्टी) draws and measures lines, a protractor (कोनमापक) measures and draws angles, a divider (विभाजक) picks up a length, and a set square (सेट-स्क्वेअर) draws 90°, 45°, 30° and 60°.

Rule 1: Tools and notation — The length of segment AB is written l(AB), for example l(AB) = 6 cm. In a circle with centre O every point on the circle is at the same distance from O. This distance is the radius (त्रिज्या), and the diameter (व्यास) is 2 × radius.

Circle with radius and diameter, a segment of length 8 cm and a 30 degree angle.Hradiusdiameter = 2 × radius0481216l(AB) = 8 cm30°Langle by protractor

In the figure : In the figure parts of a circle, a segment and an angle.

  • Easy : Easy: Draw a segment AB with l(AB) = 6 cm using a ruler.
  • Medium : Medium: A circle of radius 3 cm has diameter 2 × 3 = 6 cm.
  • Exam-type : Exam: A circle with centre O has radius 4 cm. Every point on it is 4 cm from O, and the longest chord (the diameter) is 8 cm.

Rule 2: Drawing an angle with a protractor — To draw 40°: draw the base ray LM. Place the protractor centre on L with its zero line along LM. Mark a point N at 40° on the scale that starts from 0 on LM. Join L to N. Then ∠NLM = 40°.

  • Easy : Easy: Draw ∠ABC = 50°.
  • Medium : Medium: To draw 120°, read the scale that starts with 0 on your base ray.
  • Exam-type : Exam: Draw ∠XYZ = 75°, then check with the protractor from the other arm; it must also read 75°.
Remember

Compass → circle. Ruler → line. Protractor → angle. Set square → 90°, 45°, 60°. Divider → copy length.

Common mistake

Reading the wrong scale of the protractor. Start from the scale with 0 on your base ray.

Shortcut

Memory clue: "कंपास गोल, पट्टी सरळ, कोनमापक कोन".

Spot the difference

ToolUse
CompassCircles, arcs
ProtractorMeasure, draw angles
DividerCopy a length

Practice questions

Easy
  1. 1Which tool measures angles?
  2. 2Which tool draws circles?
  3. 3A circle has radius 5 cm. Find its diameter.
Medium
  1. 4Write "the length of segment PQ is 7 cm" in symbols.
  2. 5Which angles can be drawn with a set square?
  3. 6Write the steps to draw ∠PQR = 65°.
Challenging
  1. 7A circle has diameter 14 cm. Find its radius.
  2. 8Why do we use a divider?
Show answers (try them yourself first)
  1. 1Protractor.
  2. 2Compass.
  3. 310 cm.
  4. 4l(PQ) = 7 cm.
  5. 590°, 45°, 30° and 60°.
  6. 6Draw ray QR; place the protractor centre on Q with the zero line on QR; mark 65°; join Q to that point P.
  7. 714 ÷ 2 = 7 cm.
  8. 8To pick up a length from one place and mark it on another, or to compare lengths without measuring.

Perpendicular lines

From daily life

The corner of your notebook, the junction of a wall and the floor, and the hands of a clock at 3 o'clock all make a perfect square corner. Mathematicians call these lines perpendicular.

Simple definition

Two lines that meet at 90° are perpendicular (लंब). We write PQ ⊥ RS.

Rule 1: Perpendicular lines — If PQ ⊥ RS and they cross at O, then all four angles at O are right angles: ∠POR = ∠ROQ = ∠QOS = ∠SOP = 90°.

Two lines meeting at 90 degrees: perpendicular lines.90°90°90°90°PQRSLine PQ ⊥ Line RSmeet at 90°4 right anglesCorner of a notebook,plumb line and floor

In the figure : In the figure perpendicular lines.

  • Easy : Easy: A wall and the floor of a room are perpendicular.
  • Medium : Medium: At 9 o'clock the clock hands make 90°, so they are perpendicular.
  • Exam-type : Exam: At 6 o'clock the hands make 180° (a straight line), so they are not perpendicular.

Rule 2: Checking and using perpendiculars — Place one edge of a set square along one line. If the other line runs along the second edge, the lines are perpendicular. A plumb line (a weight hanging on a string) hangs straight down, so builders use it to keep a wall perpendicular to the ground.

  • Easy : Easy: A flagpole standing straight on level ground is perpendicular to the ground.
  • Medium : Medium: Two lines meet at 85°. They are not perpendicular, because the angle must be exactly 90°.
  • Exam-type : Exam: In the letter T the stem and the bar meet at 90°, so the stem is perpendicular to the bar.
Remember

Perpendicular = 90°. Symbol ⊥. Two perpendicular lines make four right angles.

Common mistake

Calling any two crossing lines perpendicular. The angle must be exactly 90°.

Shortcut

Memory clue: "लंब = ⊥ = 90° = चौकोनी कोपरा".

Spot the difference

SituationPerpendicular?
Wall and floorYes
Clock hands at 9Yes
Clock hands at 6No

Practice questions

Easy
  1. 1What is the angle between perpendicular lines?
  2. 2Which symbol means perpendicular?
  3. 3Write "PQ is perpendicular to RS" in symbols.
Medium
  1. 4How many right angles are made where two perpendicular lines cross?
  2. 5Give two examples of perpendicular lines in your classroom.
  3. 6Are the clock hands perpendicular at 6 o'clock?
Challenging
  1. 7Two lines meet at 85°. Are they perpendicular? Why?
  2. 8Why do builders use a plumb line?
Show answers (try them yourself first)
  1. 190°.
  2. 2⊥.
  3. 3PQ ⊥ RS.
  4. 44.
  5. 5Two adjacent edges of the blackboard; the wall and the floor.
  6. 6No, they make 180°.
  7. 7No, perpendicular lines meet at exactly 90°.
  8. 8It hangs straight down, so it shows the exact vertical for building a wall perpendicular to the ground.

Drawing perpendiculars

From daily life

A carpenter has to mark a line exactly square to the edge of a plank. With a compass and ruler alone, a perpendicular can be drawn accurately, without using a set square.

Simple definition

A perpendicular can be drawn (1) at a point on a line, or (2) from a point outside the line. Both use arcs of equal radius.

Rule 1: Perpendicular at a point on the line — Let C be a point on line XY. (1) With centre C draw arcs cutting XY at E and F. (2) With a larger radius (more than half of EF), draw arcs from E and from F that meet at G. (3) Join C and G. Then CG ⊥ XY.

Drawing a perpendicular with compass and ruler: at a point on the line, and from a point outside.APQRPoint on the lineAR ⊥ PQPABQPoint outside the linePQ ⊥ AB

In the figure : In the figure a perpendicular at a point and one from an outside point.

  • Easy : Easy: Draw line XY = 8 cm and mark C on it. Use the steps to draw CG ⊥ XY.
  • Medium : Medium: The arcs from E and F must be drawn with the same radius, or G will not lie straight above C.
  • Exam-type : Exam: Why radius more than half of EF? A smaller radius gives arcs that never meet.

Rule 2: Perpendicular from a point outside the line — Let D be a point outside line XY. (1) With centre D draw an arc cutting XY at M and N. (2) With centre M and then N, using the same radius, draw arcs meeting at H on the other side of the line. (3) Join D and H. Then DH ⊥ XY.

  • Easy : Easy: Draw a line XY and a point D 4 cm above it; draw DH ⊥ XY.
  • Medium : Medium: Lines DH and XY meet at O. Then ∠DOX = ∠DOY = 90°.
  • Exam-type : Exam: The shortest distance from D to line XY is l(DO), measured along the perpendicular.
Remember

Equal arcs from two points on the line meet at the new point. Join it with the given point. Check with a protractor.

Common mistake

Changing the compass radius between the two arcs. Keep it fixed.

Shortcut

Memory clue: "दोन चाप भेटतात, मग जोडा" (two arcs meet, then join).

Spot the difference

CaseFirst arcs cut line at
Point on line (C)E and F
Point outside (D)M and N

Practice questions

Easy
  1. 1Which tool draws arcs?
  2. 2After drawing a perpendicular, how do you check it?
  3. 3What is the last step in drawing a perpendicular at a point C on a line?
Medium
  1. 4In the construction at point C, where do the first arcs cut the line?
  2. 5In the construction from outside point D, what do you do after cutting the line at M and N?
  3. 6Write the steps to draw a perpendicular at point P on line AB.
Challenging
  1. 7Why must the radius be more than half of EF?
  2. 8A student gets ∠DOX = 88° after the construction. What could have gone wrong?
Show answers (try them yourself first)
  1. 1Compass.
  2. 2Measure the angle with a protractor; it must be 90°.
  3. 3Join C and G.
  4. 4At E and F, equal distances from C.
  5. 5From M and N draw arcs of equal radius meeting at H on the other side, then join D and H.
  6. 6Arcs from P cut AB at two points; larger equal arcs from those points meet at Q; join PQ.
  7. 7If it is less, the two arcs will not meet.
  8. 8The radius probably changed between the two arcs, or the arcs were not equal.

Copying an angle

From daily life

A tailor copies a pattern piece onto fresh cloth. In geometry we copy an angle in the same way, so that the new angle is exactly equal to the old one, without a protractor.

Simple definition

To copy an angle (कोन उतरवणे) we use arcs of equal radius and one measured distance, PQ-style, with a compass.

Rule 1: Steps to copy an angle — Let ∠CBD be the given angle with vertex B. (1) Draw a ray ST. (2) With centre B draw an arc cutting BC at P and BD at Q. (3) With the same radius and centre S, draw an arc cutting ST at U. (4) Measure PQ with the compass. From U, cut the first arc at V. (5) Draw ray SV. Then ∠VSU = ∠CBD.

Copying an angle: the same radius and the gap PQ copied to KJ.MPQGiven angleAKJCopy (same radius)PQ = KJ

In the figure : In the figure copying an angle.

  • Easy : Easy: Copy any acute angle ∠CBD on to a ray ST.
  • Medium : Medium: After copying, measure both angles with the protractor; they should read the same, for example 50° and 50°.
  • Exam-type : Exam: The same method copies an obtuse angle, for example 110°.

Rule 2: Why it works — The arc radius is the same at B and at S, and the distance PQ is copied to UV. So the two corners have the same openings, and the angles are equal.

  • Easy : Easy: If the radius is changed between the two arcs, the copy will be wrong.
  • Medium : Medium: A copy of a 70° angle should measure 70°.
  • Exam-type : Exam: Two copies of the same angle on different rays are equal to each other.
Remember

Same radius at both vertices. Measure PQ with the compass and copy it.

Common mistake

Changing the compass opening after drawing the first arc. Keep it fixed until the second arc is done.

Shortcut

Memory clue: "त्रिज्या तीच, अंतर PQ तेच" (same radius, same distance PQ).

Spot the difference

StepAction
1Draw ray ST
2Arc from B cuts arms at P, Q
3Same arc from S cuts ST at U
4Copy PQ to UV, draw SV

Practice questions

Easy
  1. 1What is the first step to copy an angle?
  2. 2Which tool is used to measure PQ?
  3. 3Should the radius of the two arcs be the same or different?
Medium
  1. 4After copying, how can you check?
  2. 5Write the steps to copy ∠CBD onto ray ST.
  3. 6After copying, which angle equals ∠CBD?
Challenging
  1. 7Can an obtuse angle be copied with this method?
  2. 8A student changed the compass opening after the first arc. What happens to the copy?
Show answers (try them yourself first)
  1. 1Draw a new ray.
  2. 2The compass.
  3. 3The same.
  4. 4Measure both angles with a protractor; they must be equal.
  5. 5Arc from B cutting arms at P, Q; same arc from S cutting ST at U; copy PQ from U to cut the arc at V; draw SV.
  6. 6∠VSU.
  7. 7Yes, the steps are the same.
  8. 8The new angle will not be equal to the old angle.

Angle bisector

From daily life

Fold a sheet of paper so that the two arms of an angle fall exactly on each other. The crease cuts the angle into two equal halves. This crease is the angle bisector.

Simple definition

A ray that divides an angle into two equal parts is the angle bisector (कोनदुभाजक).

Rule 1: Bisecting with a compass — Let ∠AOB be the angle with vertex O. (1) With centre O draw an arc cutting OA at P and OB at Q. (2) With centre P and then Q, using the same radius, draw arcs meeting at C. (3) Draw ray OC. Then ∠AOC = ∠COB, each half of ∠AOB.

Bisecting an angle with a compass: ray PT divides it into two equal parts.PRDTBisector PT110° → 55° + 55°90° → 45° + 45°30° → 15° + 15°half = angle ÷ 2

In the figure : In the figure the angle bisector.

  • Easy : Easy: Bisector of 70° gives 35° and 35°.
  • Medium : Medium: Bisector of 90° gives 45° and 45°.
  • Exam-type : Exam: Bisect 60° twice: 60° → 30° → 15°.
Live figure

Angle bisector

In the app you can move this figure with your finger.

Open in the app

Rule 2: Paper folding and half angles — Fold the paper so that the arms coincide. The crease is the bisector. Half of 120° is 60°, half of 80° is 40° and half of 50° is 25°.

  • Easy : Easy: Bisector of 100° makes two angles of 50°.
  • Medium : Medium: Bisecting a 180° straight angle gives two 90° angles.
  • Exam-type : Exam: Ray YW bisects ∠XYZ = 140°. Then ∠XYW = ∠WYZ = 70°.
Remember

Bisector = equal halves. Same radius from P and Q. Check both halves with a protractor.

Common mistake

Letting the compass opening change when drawing the arcs from P and Q. Keep it fixed.

Shortcut

Memory clue: "घडी घाला, अर्धे अर्धे" (fold it, two halves).

Spot the difference

AngleEach half
120°60°
90°45°
50°25°

Practice questions

Easy
  1. 1What is the bisector of 100°?
  2. 2Bisect a 64° angle. What is each half?
  3. 3Each half of a right angle is how many degrees?
Medium
  1. 4Write the steps to bisect ∠AOB.
  2. 5Bisect a 30° angle twice. What angle do you get?
  3. 6Ray YW bisects ∠XYZ = 140°. Find ∠XYW and ∠WYZ.
Challenging
  1. 7How can paper folding bisect an angle?
  2. 8Bisecting a straight angle gives what kind of angles?
Show answers (try them yourself first)
  1. 150° each.
  2. 232°.
  3. 345°.
  4. 4Arc from O cuts OA, OB at P, Q; equal arcs from P and Q meet at C; draw ray OC.
  5. 530° → 15° → 7.5°.
  6. 6Both are 70°.
  7. 7Fold so that both arms lie exactly on each other; the crease is the bisector.
  8. 8Two right angles (90° each), so the bisector is perpendicular to the line.