GANITA PRAKASH · GRADE 8 · PART II · CHAPTER 2

Chapter Summary

Think first. Use AI purposefully. Verify always.

Concept Map

Theorem / Formula Bank

Right triangle: $a^{2}$ + $b^{2}$ = $c^{2}$ , with c opposite the established right angle.
Converse: for a triangle with longest side c, $a^{2}$ + $b^{2}$ = $c^{2}$ establishes the right angle.
Isosceles right triangle: c=a $\sqrt{2}$ . An arbitrary isosceles triangle does not satisfy this rule.
Square diagonal: d=a $\sqrt{2}$ . Midpoint-square area is $a^{2}$ /2.
1.414< $\sqrt{2}$ <1.415; finite decimals are approximations.
Primitive triple: positive integer sides satisfying the theorem, with gcd 1.
Odd-square generator: n= $\frac{(m^{2} + 1)}{2}$ , with odd m≥3, gives (n−1,m,n).

For the converse, construct a right triangle with legs a,b. Its hypotenuse has length c by the theorem. SSS makes it congruent to the given triangle, establishing the right angle.

Visual Recall

Move the pieces; preserve the area

Smaller side a (larger side b = 6): 3

Two cuts separate W and X from the five-sided piece V. Total area: $3^{2}$ + $6^{2}$ = 45.

All four new sides are congruent hypotenuses. The two acute angles of a right triangle are complementary, so each new corner is 90°. This establishes a square before using its area formula.

All eleven mathematical labs →

Triple Bank

Primitive with all entries at most 20: (3,4,5), (5,12,13), (8,15,17). Other triples within that bound: (6,8,10), (9,12,15), (12,16,20). Further examples: (7,24,25), (12,35,37), (15,36,39).

Important Connections

Congruence and area preservation establish the dissection. The theorem connects area to length, then supports exact radicals, integer triples and applications.

On a unit lattice, every square area has the form p²+q² for integers p,q not both zero. The odd-square generator produces primitive triples but does not produce them all.

Common Mistakes

Doubling the side makes four times the area, not twice.
Halving the side leaves one quarter of the area.
The formula needs an isosceles right triangle, not just any isosceles triangle.
Write √2 ≈ 1.414, never √2 = 1.414. Repeating decimals such as 1/3 are rational.
Equal total area alone does not prove the new boundary is a square; check its sides and angles.
Never use the theorem for a triangle merely because its drawing looks right-angled.
An integer right-triangle triple need not be primitive.
(8,15,17) is primitive but not generated: neither leg is 16, one less than its hypotenuse.
Use half-diagonals in a rhombus. Distinguish a square’s area from its side length.
Square the lengths before adding or subtracting.
The positive square root gives the length; negative lengths are not valid.

Mini Examples

DOUBLE — Area of the square constructed on its diagonal

Given

A square with side 3 units.

To Find

Area of the square constructed on its diagonal

Property

Four congruent half-square triangles form the new square.

Solution

The original area is $3^{2}$ = 9 square units.
The new square contains twice as many congruent triangles.

Result

The new area is 18 square units.

HALVE — Area of its midpoint square

Given

An outer square of side 8 units.

To Find

Area of its midpoint square

Property

Subtract the four congruent corner triangles.

Solution

Each corner triangle has area $\frac{(4 \times 4)}{2}$ = 8.
Outer area = 64; removed area = 4 × 8 = 32.

Result

Midpoint-square area = 32 square units.

ROOT — Length of its hypotenuse

Given

An isosceles right triangle with equal legs 12.

To Find

Length of its hypotenuse

Property

$c^{2}$ = $2 a^{2}$ from square doubling.

Solution

$c^{2}$ = 2 × $12^{2}$ = 288.
c = $\sqrt{288}$ = 12 $\sqrt{2}$ .
$16^{2}$ < 288 < $17^{2}$ , so 16 < c < 17.

Result

The hypotenuse is exactly 12 $\sqrt{2}$ units.

BOUND — Three-decimal lower and upper bounds for √2

Given

Positive candidates 1.414 and 1.415.

To Find

Three-decimal lower and upper bounds for $\sqrt{2}$

Property

Order of squares of positive numbers.

Solution

1. $414^{2}$ = 1.999396 < 2.
1. $415^{2}$ = 2.002225 > 2.

Result

1.414 < $\sqrt{2}$ < 1.415.

COMBINE — Area of the square made by the source dissection

Given

Squares with sides 3 and 4 units.

To Find

Area of the square made by the source dissection

Property

Rearrangement preserves area; the boundary is a square by congruence and complementary angles.

Solution

The two original areas are 9 and 16.
The central piece stays fixed; the two right triangles are rigidly moved.

Result

The combined square has area 25 square units.

THEOREM — Length of the other leg b

Given

A right triangle with hypotenuse 17 cm and one leg 8 cm.

To Find

Length of the other leg b

Property

$8^{2}$ + $b^{2}$ = $17^{2}$ .

Solution

$b^{2}$ = 289 − 64 = 225.
b = $\sqrt{225}$ = 15; length is positive.

Result

The other leg is 15 cm.

TRIPLES — Its primitive triple and scale factor

Given

The triple (15,36,39).

To Find

Its primitive triple and scale factor

Property

Divide every entry by their greatest common divisor.

Solution

gcd(15,36,39) = 3.
Dividing gives (5,12,13).
$5^{2}$ + $12^{2}$ = $13^{2}$ and gcd(5,12,13) = 1.

Result

Primitive triple (5,12,13), scale factor 3.

GENERATE — The triple generated from this odd square

Given

Odd square 49 = $7^{2}$ .

To Find

The triple generated from this odd square

Property

2n−1 = 49 and (n−1) $^{2}$ + 49 = $n^{2}$ .

Solution

n = 25 and n−1 = 24.
$24^{2}$ + $7^{2}$ = 576 + 49 = 625 = $25^{2}$ .

Result

The primitive triple is (24,7,25).

APPLY — Length of each rhombus side

Given

A rhombus has diagonals 24 and 70 units.

To Find

Length of each rhombus side

Property

Rhombus diagonals bisect each other at right angles.

Solution

Half-diagonals are 12 and 35.
A side is the hypotenuse: $s^{2}$ = $12^{2}$ + $35^{2}$ = 1369.
s = 37.

Result

Each side is 37 units.

5-Question Quick Check

Progress: $\frac{0}{5}$ checked correctly

Q1. A square has area 18. Find the area of the square on its diagonal.Your numerical answer

Attempts: 0

Q2. Which statement is correct?√2=1.4141.415<√21.414<√2<1.415

Attempts: 0

Q3. A right triangle has hypotenuse 17 and leg 8. Find the other leg.Your numerical answer

Attempts: 0

Q4. Why does the generator miss (8,15,17)?It is not primitiveIt is not a right triangleNeither leg is 16

Attempts: 0

Q5. Rhombus diagonals are 24 and 70. Find its side.Your numerical answer

Attempts: 0

Concept Map

Theorem / Formula Bank

Visual Recall

Move the pieces; preserve the area

Triple Bank

Important Connections

Common Mistakes

Mini Examples

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5-Question Quick Check

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