Constructions and Tilings | FIO

Question 6

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

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Solution
Understand the Question
  • A 90∘90^\circ angle can be constructed using a rope by forming a 3:4:53 : 4 : 5 right-angled triangle based on the Pythagorean triple (32+42=523^2 + 4^2 = 5^2).
  • By marking a 12-unit12\text{-unit} rope to create three connected segments of lengths 33, 44, and 55 units, the angle opposite the longest side (5 units5\text{ units}) is always exactly 90∘90^\circ.

Step 1 · Verify the 3-4-5 Triangle Property

By the converse of Pythagoras' theorem, a triangle with sides 33, 44, and 55 units is a right-angled triangle:

32+42=9+16=25\begin{aligned} 3^2 + 4^2 &= 9 + 16 \\ &= 25 \end{aligned}

52=255^2 = 25

Since 32+42=523^2 + 4^2 = 5^2, the angle opposite the longest side (5 units5\text{ units}) is 90∘90^\circ.Diagram 1

Step 2 · Mark the Rope and Form Triangle ABC

Mark points on a rope at 00, 33, 88, and 12 units12\text{ units} to get segments of lengths 33, 55, and 44 units.

To construct a 90∘90^\circ angle at point A\text{A} on line XY\text{XY}:

  1. Fasten both the 0-unit0\text{-unit} and 12-unit12\text{-unit} marks to a pole at point A\text{A}.
  2. Stretch the 3-unit3\text{-unit} segment along line XY\text{XY} and fix it with a pole at point B\text{B}.
  3. Pull the 8-unit8\text{-unit} mark away from line XY\text{XY} until both remaining sides are taut, and fix it at point C\text{C}.Diagram 2

The side lengths of ΔABC\Delta \text{ABC} are: AB=3−0=3 units\text{AB} = 3 - 0 = 3\text{ units} BC=8−3=5 units\text{BC} = 8 - 3 = 5\text{ units} CA=12−8=4 units\text{CA} = 12 - 8 = 4\text{ units}

Step 3 · Identify the 90-Degree Angle

In ΔABC\Delta \text{ABC}, the longest side is BC=5 units\text{BC} = 5\text{ units}.

The angle opposite to side BC\text{BC} is ∠BAC\angle \text{BAC} at vertex A\text{A}.

∠BAC=90∘\angle \text{BAC} = 90^\circ

Therefore, segment AC\text{AC} is perpendicular to line XY\text{XY} at point A\text{A}.

Answer

A 90∘90^\circ angle is constructed at point A\text{A} by forming a 3-4-53\text{-}4\text{-}5 triangle using a 12-unit12\text{-unit} rope loop marked at 00, 33, 88, and 12 units12\text{ units}, where the angle ∠BAC\angle \text{BAC} opposite the 5-unit5\text{-unit} side is 90∘90^\circ.

Common Mistakes
  • Wrong Vertex for 90∘90^\circ: The right angle is formed opposite the longest side (5 units5\text{ units}), which is at point A\text{A} (between the 3-unit3\text{-unit} and 4-unit4\text{-unit} sides), not at B\text{B} or C\text{C}.
  • Incorrect Marking: Marking the rope at equal intervals instead of cumulative distances (00, 33, 88, 1212) will fail to produce sides of lengths 33, 55, and 44.
  • Loose Rope: The rope must be pulled completely taut at all three vertices; any slack will distort the side lengths and the angle will not be exactly 90∘90^\circ.

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

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Q8

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Q9

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Q10

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Q11

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Q12

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Q13

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Q14

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Q15

Construct 4 pairs of parallel lines in different orientations.

Q16

Construct the following figure.

Q17

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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

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Q24

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