Constructions and Tilings | FIO

Question 7

Construct at least 4 different angles. Draw their bisectors.

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Solution
Understand the Question

An angle bisector is a ray that divides an angle into two equal parts.

To bisect any angle with vertex VV and arms:

  1. Draw an arc with center at the vertex VV cutting both arms at two points.
  2. From these two intersection points, draw two intersecting arcs inside the angle using the same compass width.
  3. Join vertex VV to the point of intersection of these arcs to form the bisector ray.

We construct four different types of angles: acute, obtuse, right, and straight angle, and draw their bisectors.

Step 1 · Construct and Bisect Acute Angle ∠ABC\angle ABC

Draw an acute angle ∠ABC\angle ABC.Diagram 1

  1. Place the compass at vertex BB and draw an arc cutting arm BA\text{BA} at A′A' and arm BC\text{BC} at C′C'.
  2. With center A′A', draw an arc in the interior of the angle.
  3. With the same radius and center C′C', draw another arc cutting the previous arc at B′B'.
  4. Draw the ray from BB through B′B'.

Ray BB′\text{BB}' is the bisector of ∠ABC\angle ABC.

Step 2 · Construct and Bisect Obtuse Angle ∠DEF\angle DEF

Draw an obtuse angle ∠DEF\angle DEF.

  1. Place the compass at vertex EE and draw an arc cutting arm ED\text{ED} at D′D' and arm EF\text{EF} at F′F'.
  2. With center D′D', draw an arc inside the angle.
  3. With the same radius and center F′F', draw another arc cutting the previous arc at E′E'.
  4. Draw the ray from EE through E′E'.

Ray EE′\text{EE}' is the bisector of ∠DEF\angle DEF.

Step 3 · Construct and Bisect Right Angle ∠GHI\angle GHI

Draw a right angle ∠GHI\angle GHI (90∘90^\circ).

  1. Place the compass at vertex HH and draw an arc cutting arm HG\text{HG} at G′G' and arm HI\text{HI} at I′I'.
  2. With center G′G', draw an arc inside the angle.
  3. With the same radius and center I′I', draw another arc cutting the previous arc at H′H'.
  4. Draw the ray from HH through H′H'.

Ray HH′\text{HH}' is the bisector of ∠GHI\angle GHI.

Step 4 · Construct and Bisect Straight Angle ∠JKL\angle JKL

Draw a straight angle ∠JKL\angle JKL (180∘180^\circ).

  1. Place the compass at vertex KK and draw an arc cutting arm KJ\text{KJ} at J′J' and arm KL\text{KL} at L′L'.
  2. With center J′J', draw an arc above the line.
  3. With the same radius and center L′L', draw another arc cutting the previous arc at K′K'.
  4. Draw the ray from KK through K′K'.

Ray KK′\text{KK}' is the bisector of ∠JKL\angle JKL.

Answer

The bisectors of the four constructed angles are ray BB′\text{BB}' for ∠ABC\angle ABC, ray EE′\text{EE}' for ∠DEF\angle DEF, ray HH′\text{HH}' for ∠GHI\angle GHI, and ray KK′\text{KK}' for ∠JKL\angle JKL.

Common Mistakes
  • Changing Compass Radius: Changing the compass width between drawing the intersecting arcs in the interior of the angle will lead to an inaccurate bisector.
  • Centering Arcs Incorrectly: Drawing the interior arcs from the vertex instead of the points of intersection on the arms (A′A' and C′C').
  • Small Radius: Taking a radius less than half the distance between the two arm intersection points, causing the interior arcs not to intersect.

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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