Chapter 2 · Practice set 2Textbook pages 29–30

Practice Set 2 – Animated Solutions കൂട്ടലും കുറയ്ക്കലും

Four questions about perimeters that contain square roots. The trick in every one: keep √2, √3, … as they are while you add and subtract, cancel what cancels, and only at the very end turn what is left into a decimal.

📖 Tools you will use (pages 3, 6 and 8 of this guide)

Pythagoras: hypotenuse² = (side)² + (side)², so a missing side is found from areas of squares.
Decimals: √2 = 1.41421…, √3 = 1.73205…, √5 = 2.23606…, √10 = 3.16227…, √11 = 3.31662…
Adding and subtracting: (1 + √2 + √3) − (2 + √2) = √3 − 1 – the √2 cancels before any decimals are needed.
Rounding: correct to a centimetre = 2 decimal places of a metre; to a millimetre = 3.

Question 1 – hypotenuse 1½ m, one side ½ m

Q1The hypotenuse of a right triangle is 1½ metres and one side is ½ metre. Calculate its perimeter correct to a centimetre.

ഒരു മട്ടത്രികോണത്തിന്റെ കർണ്ണം 1½ മീറ്ററും, മറ്റൊരു വശം ½ മീറ്ററുമാണ്. അതിന്റെ ചുറ്റളവ്, സെന്റിമീറ്റർ വരെ കൃത്യമായി കണക്കാക്കുക.

Square the two known sides: (1½)² = 2¼ and (½)² = ¼. The square on the third side is the difference.

🎬 Solution animation

Answer: third side² = 2¼ − ¼ = 2, so the third side is √2 m. Perimeter = 1½ + ½ + √2 = 2 + √2 ≈ 3.414 m ≈ 3.41 m (341 cm) correct to a centimetre.

🌍 Why this matters

The sides are "nice" fractions, yet the third side is √2. Perimeters of real plots and frames often mix fractions and square roots like this; keep the root until the end and round once.

Question 2 – half of an equilateral triangle

Q2An equilateral triangle with sides 2 metres is cut through a vertex into two equal halves. (i) What is the perimeter of one half? (ii) How much less is it than the perimeter of the whole triangle?

The cut is the height. Each half is a right triangle with hypotenuse 2 m and base 1 m, so its height is √(4 − 1) = √3 m.

🎬 Solution animation

Answer: each half has sides 1 m, 2 m and √3 m. (i) Perimeter = 1 + 2 + √3 = 3 + √3 ≈ 4.73 m. (ii) The whole triangle has perimeter 6 m, so the half is less by 6 − (3 + √3) = 3 − √3 ≈ 1.27 m.

💡 Notice

Half the triangle does not have half the perimeter: 4.73 m is much more than 3 m, because the new cut edge √3 is added to it.

Question 3 – the tenth triangle of the fan

Q3Right triangles are drawn one after another as in the figure, each with one side 1 m and the hypotenuse of each becoming a side of the next. (i) What are the sides of the tenth triangle? (ii) How much more is the perimeter of the tenth triangle than that of the ninth?

Hypotenuse² of the first is 2, of the second 3, of the third 4… So the nth triangle has sides 1, √n and √(n + 1).

🎬 Solution animation

Answer: (i) the tenth triangle has sides 1 m, √10 m ≈ 3.162 m and √11 m ≈ 3.317 m. (ii) (1 + √10 + √11) − (1 + √9 + √10) = √11 − 3 ≈ 3.317 − 3 = 0.317 m (about 31.7 cm).

Question 4 – sides √2 cm and √3 cm

Q4The perpendicular sides of a right triangle are √2 cm and √3 cm. What is the length of the hypotenuse? How much more is the sum of the perpendicular sides than the hypotenuse?

The squares on the perpendicular sides have areas 2 and 3. Add them.

🎬 Solution animation

Answer: hypotenuse² = 2 + 3 = 5, so the hypotenuse is √5 cm ≈ 2.236 cm. The perpendicular sides together are √2 + √3 ≈ 3.146 cm, which is more by √2 + √3 − √5 ≈ 0.910 cm.

💡 A warning hidden in this question

√2 + √3 is not √5. Adding the numbers under the roots gives the hypotenuse, not the sum of the sides – and the two sides together are always longer than the third (page 8).

Make your own: legs √a and √b

🎚️ Choose the areas of the squares on the two perpendicular sides

a b

Check yourself

🧭 Summary

Find a missing side with areas of squares; it is often a square root.
Write perimeters exactly (2 + √2, 3 + √3, 1 + √10 + √11) and subtract exactly – roots that appear in both cancel.
Turn the final answer into a decimal and round to what is asked (cm = 2 places, mm = 3 places).

Next: Appendix – why no fraction squares to 2 ▶