Chapter 9 · Practice sets 3 & 4Textbook page 176

Practice Sets 3 & 4 – Solutions മധ്യബിന്ദു

Two tools do all the work here: the midpoint of x and y is ½(x + y), and |x − a| = |x − b| means x is equally far from a and b – so it is their midpoint.

📖 What you need (page 10 of this guide)

Midpoint of x and y = ½(x + y), whatever their signs.
|x − a| is the distance between x and a; |x + b| = |x − (−b)|.

Practice set 3 · Question 1 · Eight midpoints

Q1. Find the number of the midpoint of each pair on the number line: (i) 1, −5 (ii) −1, −5 (iii) −½, −⅓ (iv) −½, ¾ (v) −2.5, 3.5 (vi) 1.3, 8.7 (vii) −√2, −√3 (viii) −√3, √7 സംഖ്യാരേഖയിൽ, ചുവടെ പറഞ്ഞിരിക്കുന്ന ഓരോ ജോടി സംഖ്യകളും സൂചിപ്പിക്കുന്ന ബിന്ദുക്കളുടെ മധ്യബിന്ദുവിനെ സൂചിപ്പിക്കുന്ന സംഖ്യ കണക്കാക്കുക.

Add the two numbers (careful with signs), then halve. For fractions use a common denominator: −½ + (−⅓) = −3/6 − 2/6 = −5/6.

🎬 Solution animation

Answer

(i) −2(ii) −3(iii) −5/12(iv) ⅛(v) 0.5(vi) 5(vii) −½(√2 + √3) ≈ −1.573(viii) ½(√7 − √3) ≈ 0.457

💡 Irrational midpoints

The rule works for irrational numbers too: the midpoint of −√2 and −√3 is ½(−√2 − √3) = −½(√2 + √3). A decimal is only an approximation; the exact answer keeps the roots.

Practice set 3 · Question 2 · Four equal parts

Q2. Find the numbers of the points that divide the part of the number line between 1 and 2 into four equal parts.

The midpoint of 1 and 2 cuts it into two halves. Then take the midpoint of each half.

🎬 Solution animation

Answer

1¼, 1½, 1¾ (each part is ¼ long).

Practice set 4 · Equal-distance equations

Q. Find x in each equation: (i) |x − 1| = |x − 3| (ii) |x − 3| = |x − 4| (iii) |x + 2| = |x − 5| (iv) |x| = |x + 1| ചുവടെയുള്ള ഓരോ സമവാക്യവും ശരിയാകുന്ന x കണക്കാക്കുക.

Each side is a distance from x to a fixed number. Write |x + 2| as |x − (−2)| and |x| as |x − 0|. Equal distances mean x is the midpoint of the two fixed numbers.

🎬 Solution animation

Answer

(i) x = 2(ii) x = 3½(iii) x = 1½(iv) x = −½

🌍 Why this matters

"Equally far from two places" is how a shop owner picks a spot halfway between two villages along a road, and how a referee finds the centre spot between two goal lines. In geometry the same idea gives the perpendicular bisector of a line segment.

Solve your own

🎚️ |x − a| = |x − b|

a b

Check yourself

🧭 Summary

Midpoint = ½(x + y): add, then halve – works for fractions, decimals and irrationals.
Repeated midpoints divide a part of the line into 2, 4, 8 … equal parts.
|x − a| = |x − b| (a ≠ b) has exactly one solution: x = ½(a + b).

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