Constructions and Tilings | FIO

Question 13

Construct at least 4 different angles in different orientations without taking any measurement. Make a copy of all these angles.

Question diagram 1
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Solution
Understand the Question
  • To copy an angle without using a protractor (measurement), we use a compass and straightedge (ruler).
  • The method relies on transferring equal arc radii from both vertices and then transferring the exact arc width (chord length between the arms) to construct an identical angle in any orientation.

Step 1 · Copy the First Angle ∠ABC\angle ABC

Let the given angle be ∠ABC\angle ABC with vertex BB.Diagram 1

Steps of Construction:

  1. Draw a ray B′A′B'A' to form one arm of the new angle.
  2. With vertex BB as centre and any convenient radius, draw an arc intersecting arm BABA at MM and arm BCBC at NN.
  3. With the same radius and centre B′B', draw an arc intersecting ray B′A′B'A' at M′M'.
  4. Set the compass opening equal to the distance MNMN.
  5. With M′M' as centre and radius MNMN, draw an arc intersecting the previous arc at N′N'.
  6. Draw ray B′C′B'C' passing through N′N'.

Thus, ∠A′B′C′\angle A'B'C' is an exact copy of ∠ABC\angle ABC.

Step 2 · Copy the Second Angle ∠DEF\angle DEF

Let the second given angle in a different orientation be ∠DEF\angle DEF with vertex EE.Diagram 2

Steps of Construction:

  1. Draw a ray E′D′E'D' as one arm of the copied angle.
  2. With centre EE and a convenient radius, draw an arc cutting arm EDED at OO and arm EFEF at PP.
  3. With the same radius and centre E′E', draw an arc cutting ray E′D′E'D' at O′O'.
  4. Adjust compass width to measure distance OPOP.
  5. With centre O′O' and radius OPOP, draw an arc intersecting the first arc at P′P'.
  6. Draw ray E′F′E'F' through P′P'.

Thus, ∠D′E′F′\angle D'E'F' is an exact copy of ∠DEF\angle DEF.

Step 3 · Copy the Third Angle ∠GHI\angle GHI

Let the third given angle in another orientation be ∠GHI\angle GHI with vertex HH.Diagram 3

Steps of Construction:

  1. Draw a ray H′G′H'G' as one arm of the new angle.
  2. With centre HH and a convenient radius, draw an arc cutting arm HGHG at QQ and arm HIHI at RR.
  3. With the same radius and centre H′H', draw an arc cutting ray H′G′H'G' at Q′Q'.
  4. Adjust compass width to measure distance QRQR.
  5. With centre Q′Q' and radius QRQR, draw an arc intersecting the previous arc at R′R'.
  6. Draw ray H′I′H'I' passing through R′R'.

Thus, ∠G′H′I′\angle G'H'I' is an exact copy of ∠GHI\angle GHI.

Step 4 · Copy the Fourth Angle ∠JKL\angle JKL

Let the fourth given angle in a different orientation be ∠JKL\angle JKL with vertex KK.Diagram 4

Steps of Construction:

  1. Draw a ray K′J′K'J' as one arm of the new angle.
  2. With centre KK and a convenient radius, draw an arc cutting arm KJKJ at SS and arm KLKL at TT.
  3. With the same radius and centre K′K', draw an arc cutting ray K′J′K'J' at S′S'.
  4. Set compass opening equal to distance STST.
  5. With centre S′S' and radius STST, draw an arc cutting the previous arc at T′T'.
  6. Draw ray K′L′K'L' through T′T'.

Thus, ∠J′K′L′\angle J'K'L' is an exact copy of ∠JKL\angle JKL.

Answer

Four angles in different orientations are copied:

  • ∠A′B′C′=∠ABC\angle A'B'C' = \angle ABC
  • ∠D′E′F′=∠DEF\angle D'E'F' = \angle DEF
  • ∠G′H′I′=∠GHI\angle G'H'I' = \angle GHI
  • ∠J′K′L′=∠JKL\angle J'K'L' = \angle JKL
Common Mistakes
  • Changing the Compass Radius: Altering the compass width between drawing the initial arc on the original angle and the new ray will distort the angle size.
  • Incorrect Arc Width Measurement: Measuring the distance between arms from points other than the exact intersection points of the original arc.

More questions in FIO

Q1

When constructing the perpendicular bisector, is it necessary to have the same radius for the arcs above and below XYXY? Explore this through construction, and then justify your answer.

[Hint 1: Any point that is of the same distance from XX and YY lies on the perpendicular bisector.

Hint 2: We can draw the whole line if any two of its points are known.]

Q2

Is it necessary to construct the pairs of arcs above and below XYXY? Instead, can we construct both the pairs of arcs on the same side of XYXY? Explore this through construction, and then justify your answer.

Q3

While constructing one pair of intersecting arcs, is it necessary that we use the same radii for both of them ? Explore this through construction, and then justify your answer.

Q4

Recreate this design using only a ruler and compass —

Q5

Justify why AB\text{AB} in Fig. 6.4 is the perpendicular bisector.

Q6

Can you think of different methods to construct a 90∘90^\circ angle at a given point on a line using a rope?

Q7

Construct at least 4 different angles. Draw their bisectors.

Q8

Construct the 8-petalled figure shown in Fig. 6.5.

Q9

In Step 2 of angle bisection, if arcs of equal radius are drawn on the other side, as shown in the figure, will the line OCOC still be an angle bisector? Explore this through construction, and then justify your answer.

Q10

What are the other angles that can be constructed using angle bisection? Can you construct 65.5∘65.5^\circ angle?

Q11

Come up with a method to construct the angle bisector using a rope.

Q12

Construct the following figure.

How do we construct the petals so that they are of the maximum possible size within a given square?

Q13

Construct at least 4 different angles in different orientations without taking any measurement. Make a copy of all these angles.

Q14

Construct the Fig. 6.6.

Q15

Construct 4 pairs of parallel lines in different orientations.

Q16

Construct the following figure.

Q17

Use support lines in Fig. 6.11 to construct a pointed arch. Make different arches, by changing the radius of the arcs.

Q18

Make your own arch designs.

Q19

Construct the following figures:

Q20

Optical Illusion: Do you notice anything interesting about the following figure? How does this happen? Recreate this in your notebook.

Q21

Construct this figure.

[Hint: Find the angles in this figure.]

Q22

Draw a line ll and mark a point PP anywhere outside the line. Construct a perpendicular to the given line ll through PP.

[Hint: Find a line segment on ll whose perpendicular bisector passes through PP.]

Q23

How can the tangram pieces be rearranged to form each of the following figures?

Q24

Are the following tilings possible?

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