Question 6

We can make a 90-degree angle. We use a special triangle. Its sides are in the ratio 3:4:5.

Step 1 — The Special 3-4-5 Triangle

Let us think about triangles. Some triangles have a special property. If a triangle has sides of length 3, 4, and 5 units, it is a right-angled triangle. The angle opposite the longest side is always 90 degrees. This is because of Pythagoras' theorem. It says the longest side squared. This equals the sum of the other two sides squared.

Let us check this rule. $3^2 + 4^2$ $= 9 + 16$ $= 25$

Now, let us check the square of the longest side. $5^2$ $= 25$ Since $3^2 + 4^2 = 5^2$, this triangle must have a 90-degree angle. This angle is opposite the side of length 5.

Step 2 — Preparing Our Rope

We need a rope for this construction. We will mark specific points on it. This helps us create the correct side lengths.

The rope has marks at 0, 3, 8, and 12 units. The first mark is at 0 units. This is the start. The second mark is at 3 units from the start. The third mark is at 8 units from the start. The fourth mark is at 12 units from the start. This is the end.

Step 3 — Setting Up the Triangle

Now we will use our marked rope. We want to make a 90-degree angle at a point on a line.

Draw a straight line. Let us call this line XY. Choose any point on line XY. Let us call this point A. We want our 90-degree angle here. Place a small pole at point A. Take the rope. Attach the 0-unit mark of the rope to the pole at A. Also, attach the 12-unit mark of the rope to the same pole at A. This makes a loop of rope. The total length of the loop is 12 units. Now, find the 3-unit mark on the rope. Stretch this part of the rope along the line XY, starting from A. Place another small pole at this 3-unit mark. Let us call this point B. So, the length of the segment AB is 3 units.

Next, find the 8-unit mark on the rope. Hold this 8-unit mark. Pull it away from the line XY. Make sure all parts of the rope are stretched tight. Place a third small pole at this 8-unit mark. Let us call this point C. We have now formed a triangle. Its corners are A, B, and C.

Let us find the lengths of the sides of triangle ABC. Side AB is from the 0-unit mark to the 3-unit mark. $AB = 3 - 0$

$\boxed{3 \text{ units}}$

Side BC is from the 3-unit mark to the 8-unit mark. $BC = 8 - 3$

$\boxed{5 \text{ units}}$

Side CA is from the 8-unit mark to the 12-unit mark. The 12-unit mark is also at A. $CA = 12 - 8$

$\boxed{4 \text{ units}}$

So, our triangle ABC has sides of lengths 3 units, 5 units, and 4 units.

Step 4 — Finding the 90-Degree Angle

We have created a triangle ABC. Its sides are 3, 5, and 4 units. From Step 1, we know about the special 3-4-5 triangle. The longest side in our triangle ABC is BC. Its length is 5 units. The angle opposite to the longest side is the 90-degree angle. The angle opposite to side BC is the angle at vertex A. This is angle BAC. Therefore, angle BAC is a 90-degree angle. This means the line segment AC is perpendicular to the line XY at point A.

Answer

(i) The method uses a principle. A triangle with sides 3:4:5 has a 90-degree angle. (ii) The rope has marks at 0, 3, 8, and 12 units. This creates segments of 3, 5, and 4 units. (iii) The 90-degree angle forms at point A. It is opposite the 5-unit side of the triangle.