Exercise 1 – Animated Solutions ഗുണനം
Six questions, all built on two ideas from Section 1: a square of area A has side √A, and √x × √y = √(xy) (so √18 = √9 × √2 = 3√2). For each question, read it, try it in your notebook (use the hint if you are stuck), then press Next ▶ to watch the solution grow step by step.
- Side of a square = √(area). A square of area 27 sq cm has side √27 cm.
- Simplify: pull out perfect squares – √27 = √(9 × 3) = 3√3, √12 = 2√3, √18 = 3√2, √50 = 5√2.
- Like terms: 3√3 + 2√3 = 5√3 (just like 3a + 2a = 5a), but √2 + √3 stays as it is.
- Equilateral triangle of side 2 cm: height = √(2² − 1²) = √3 cm (page 4 of this guide).
- Pythagoras: (hypotenuse)² = (leg)² + (leg)²; and the diagonal of a square of side a is a√2.
Question 1 · A rectangle from four equilateral triangles
Four equilateral triangles of the same size: two are cut lengthwise into halves, two are kept whole, and all the pieces are put together to make a rectangle. If every side of the triangles is 2 cm, what are the perimeter and the area of the rectangle?
(സമഭുജത്രികോണങ്ങളുടെയെല്ലാം വശങ്ങളുടെ നീളം 2 സെന്റിമീറ്റർ ആണെങ്കിൽ, ചതുരത്തിന്റെ ചുറ്റളവും പരപ്പളവും എത്രയാണ്?)
The rectangle's short side is one triangle side. Its long side is made of two triangle heights. Find the height first (half-side 1 cm, Pythagoras).
Tilers do exactly this: triangular tiles are cut in half to fill the straight edges of a floor. Knowing that a triangle of side 2 is √3 tall lets them work out how many tiles a 2 m × 3.46 m strip needs without measuring.
Question 2 · A trapezium from a square and a triangle
A square and an equilateral triangle whose sides are twice the square's side are cut and rearranged as shown to make a trapezium (ലംബകം). If the side of the square is 2 cm, what are the perimeter and the area of the trapezium?
Cut the square along a diagonal and the triangle along its height. The two triangle-halves form a rectangle; the square-halves go at the two ends. Cutting and moving pieces never changes the total area!
The perimeter mixes √3 and √2. They are not like terms, so 4√3 + 4√2 cannot be shortened – just as 4a + 4b cannot. Only a decimal (6.93 + 5.66 = 12.59) combines them, and then the answer is no longer exact.
Question 3 · Two squares side by side
The figure shows two squares placed side by side, of areas 27 sq cm and 12 sq cm. Calculate the exact length of the bottom side of the figure.
Side = √(area). Simplify √27 and √12 – both contain the same √3, so they can be added like terms.
After simplifying, the two sides add up to a single term only when the same number is left under the root (√27 = 3√3 and √12 = 2√3). Try 18 and 8, or 20 and 45, or 50 and 2. With 2 and 3 you are stuck with √2 + √3.
Question 4 · Two squares at a corner
The figure shows two squares of sides 3 cm and 2 cm joined at a corner. Find the length of the slanted line.
The slanted line is made of the two diagonals. The diagonal of a square of side a is √(a² + a²) = a√2.
Question 5 · Squares on the sides of a right triangle
In the figure, squares are drawn on two sides of a right triangle: one of area 18 sq cm on a perpendicular side, and one of area 50 sq cm on the hypotenuse. Calculate the length of the third side and the perimeter of the triangle.
The square of area 50 sits on the hypotenuse (the side opposite the right angle). Pythagoras: (other leg)² = 50 − 18. Then simplify all three square roots – they all contain √2.
Surveyors and carpenters check right angles with 3-4-5 triangles. Multiplying by √2 (or by any number) keeps the angles the same – you met this in the chapter on similar triangles.
Question 6 · Which products are whole numbers or fractions?
The product of some of these pairs of numbers is a natural number or a fraction. Find them: (i) √3, √12 (ii) √3, √1.2 (iii) √5, √8 (iv) √0.5, √8 (v) √7½, √3⅓ (vi) √(2/5), √(1/10).
Use √x × √y = √(xy): multiply the numbers under the roots. The product is a natural number or a fraction exactly when xy is a perfect square (like 36, 4, 25, 1/25).
Picture each product as the area of a rectangle with sides √x and √y. The area is √(xy) – a nice number only when xy is a perfect square.
Check yourself
- Cutting and rearranging never changes area – use it to turn triangles into rectangles and trapeziums.
- Side of a square = √(area); diagonal of a square = side × √2; height of an equilateral triangle = half-side × √3.
- Always simplify: √27 + √12 = 3√3 + 2√3 = 5√3. Different roots (√3 and √2) cannot be combined.
- √x × √y is a whole number or fraction exactly when xy is a perfect square.
Next: Section 2 – Division ▶