Chapter 5 · RevisionEverything on one page

Summary, Formula Sheet & Quick Quiz പുനരവലോകനം

All the key rules of the chapter, the formulas you will use in problems, the mistakes to avoid, flashcards for the Malayalam terms, and a scored quiz to find out what needs another look.

1. The key rules

Key 1 · p. 87 x, y എന്ന ഏതു രണ്ട് അധിസംഖ്യകളെടുത്താലും √x × √y = √(xy) √x × √y = √(xy)

Multiply under the root. √2 × √3 = √6; √5 × √20 = √100 = 10.

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Key 2 · p. 89 √x × √y = √z ആയാൽ √z/√x = √y, √z/√y = √x √z ÷ √x = √y

Division undoes multiplication. √6 ÷ √2 = √3; √50 ÷ √2 = √25 = 5.

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Key 3 · p. 90 x, y എന്ന ഏതു രണ്ട് അധിസംഖ്യകളെടുത്താലും √(x/y) = √x/√y √(x ÷ y) = √x ÷ √y

Root of a fraction. √(9/4) = 3/2; √(1/2) = √2/√4 = √2/2.

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Key 4 · p. 91 x ഏത് അധിസംഖ്യയായാലും 1/√x = √x/x 1 ÷ √x = √x ÷ x

Rationalising. 1/√2 = √2/2 ≈ 0.707; 1/√3 = √3/3.

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Key 5 · p. 93 സമഭുജത്രികോണത്തിന്റെ ഉയരം, വശത്തിന്റെ പകുതിയുടെ √3 മടങ്ങും, പരപ്പളവ്, വശത്തിന്റെ പകുതിയുടെ വർഗത്തിന്റെ √3 മടങ്ങുമാണ്. h = √3 · (a/2), area = √3 · (a/2)²

Equilateral triangle of side a: height (√3/2)a, area (√3/4)a². Side 8 ⇒ height 4√3, area 16√3.

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Key 6 · p. 95 s = ½(a + b + c) ആയാൽ പരപ്പളവ് = √(s(s−a)(s−b)(s−c)) area = √(s(s−a)(s−b)(s−c))

Heron's formula – any triangle from its three sides. 8, 7, 3 ⇒ s = 9 ⇒ √(9·1·2·6) = 6√3.

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2. Simplified square roots – click any root

These are the square roots up to √200 that can be simplified. Click one to see the perfect square hiding inside. The ones not listed (√2, √3, √5, √6, √7, √10, …) are already as simple as they get.

√18 = √(9 × 2) = √9 × √2 = 3√2 ≈ 4.243

3. Formula sheet

Situation	Formula / fact	Where
Writing multiples	4√2 = √2 + √2 + √2 + √2; ½√2 = √2/2	p. 5
Simplifying a root	√(k²m) = k√m: √18 = 3√2, √50 = 5√2, √72 = 6√2, √108 = 6√3	p. 5
Product	√x × √y = √(xy); (√x)² = x; √x × √x = x	p. 5
Quotient	√x ÷ √y = √(x/y); √(x/y) = √x/√y	p. 7
Rationalising	1/√x = √x/x; 1/(√a + b) = (√a − b)/(a − b²) using (x+y)(x−y) = x² − y²	p. 7, p. 8
Decimals to remember	√2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √6 ≈ 2.449, √7 ≈ 2.646, √10 ≈ 3.162	p. 1
Equilateral triangle, side a	height = (√3/2)a, area = (√3/4)a², perimeter 3a	p. 9
Isosceles triangle, equal sides a, base b	height = √(a² − (b/2)²), area = ½ b × height	p. 9
Regular hexagon, side a	6 equilateral triangles: area = 6 × (√3/4)a² = (3√3/2)a²; distance between parallel sides = √3·a	p. 10
Any triangle, sides a, b, c	s = ½(a + b + c); area = √(s(s−a)(s−b)(s−c))	p. 9
Diagonal of a square, side a	a√2 (e.g. 30 cm tile ⇒ 30√2 ≈ 42.4 cm)	p. 11
Right triangle	hypotenuse = √(a² + b²); squares on the legs: areas add up	p. 4

4. Common mistakes

⚠️ Watch out for these

Adding under the root. √2 + √3 is NOT √5 (1.414 + 1.732 = 3.146, but √5 = 2.236). The rule √x × √y = √(xy) is for multiplication only; √(a + b) ≠ √a + √b.
Forgetting to simplify. √72 is correct but 6√2 is the expected answer. Always hunt for the largest perfect-square factor.
Rounding too early. Keep √ until the last line: 12√5 then 26.8, not 12 × 2.2 = 26.4.
The wrong "half". Height of an equilateral triangle is √3 × half the side: side 8 ⇒ 4√3, not 8√3.
Heron's s is half the perimeter, not the whole perimeter: sides 7, 8, 9 ⇒ s = 12, not 24.
√2 × √2 = 2, not √2 or 4. And (√5 + 2)(√5 − 2) = 5 − 4 = 1, not 25 − 4.
Units. Lengths in cm, areas in sq cm. 4√3 cm is a length; 4√3 sq cm is an area.

5. Flashcards പദങ്ങൾ

Click a card to flip it.

6. Scored quiz (10 questions)

Answer every question, then press Submit. Each wrong answer tells you which page to revisit. Use √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236 for decimals.

7. Your study checklist

Pages you have marked as completed are ticked. Aim to tick them all before the model test.

🧭 Ready for the test?

If you scored 8 or more in the quiz and the checklist is complete, go to the Model Test Paper. Attempt it in 1½ hours without looking at the solutions.

Next: Model Test Paper ▶