If in a triangle, the square on the greatest side is equal to the sum of squares on the other two sides, then the triangle is right angled triangle.

**Converse
of Pythagoras Theorem**

If in a triangle,
the square on the greatest side is equal to the sum of squares on the other two
sides, then the triangle is right angled triangle.

**Example:**

In the triangle
ABC,

*AB*^{2}* + AC*^{2}* = *11^{2}* + *60^{2}* = *3721 =
61^{2}* = BC*^{2}

Hence, ∆*ABC* is a right angled triangle.

(i) There are special sets of numbers a, b and c that makes the Pythagorean
relationship true and these sets of special numbers are called Pythagorean triplets.
Example: (3, 4, 5) is a Pythagorean triplet.

(ii) Let* k *be any positive
integer greater than 1 and (a, b, c) be a Pythagorean triplet, then (ka, kb, kc)
is also a Pythagorean triplet.

Examples:

So, it is clear that we can have infinitely many Pythagorean triplets
just by multiplying any Pythagorean triplet by **k**.

We shall
now see a few examples on the use of Pythagoras theorem in problems.

** **

__Example 5.11__

In the figure,
AB ⊥ AC

a) What type
of ∅ is ABC?

b) What are
AB and AC of the ∆ABC ?

c) What is
CB called as ?

d) If AC
= AB then, what is the measure of ∠ B and ∠C ?

*Solution:*

a) ∆ABC is
right angled as AB ⊥
AC at A.

b) AB and
AC are legs of ∆ ABC .

c) CB is
called as the hypotenuse.

d) ∠ B + ∠ C = 90º and equal angles are opposite
to equal sides. Hence, ∠
B = ∠ C =90º /2 =45º

** **

__Example 5.12__

Can a right
triangle have sides that measure 5c*m*,
12c*m* and 13c*m?*

*Solution:*

Take *a* = 5 , *b* = 12 and *c* =
13

Now, *a* ^{2} +
*b*^{2} =
5^{2} + 12^{2} = 25 + 144 =
169 =
13^{2} = *c*^{2}

By the converse
of Pythagoras theorem, the triangle with given measures is a right angled triangle.

** **

__Example 5.13__

A 20-feet
ladder leans against a wall at height of 16 feet from the ground. How far is the
base of the ladder from the wall?

*Solution:*

The ladder,
wall and the ground form a right triangle with the ladder as the hypotenuse. From
the figure, by Pythagoras theorem,

20^{2}
=
16^{2} + *x*^{2}

⇒
400 =
256 +
*x*^{2}

⇒* x*^{2}* *=* *400* *−* *256* *=* *144* *=* *12^{2}* *

⇒* x *=12* *feet

Therefore,
the base (foot) of the ladder is 12 feet away from the wall.

** **

**Activity**

**1. We can construct sets
of Pythagorean triplets as follows.**

**Let m and n be any two
positive integers (m > n):**

**(a, b, c) is a Pythagorean
triplet if a **

**Complete the table.**

**2. Find all
integer-sided right angled triangles with hypotenuse 85.**

**Solution:**

(*x + y*)^{2} − 2*xy* = 85^{2}

Pythagorean triplets with hypotenuse 85.

13, 84, 85

36, 77, 85

40, 75, 85

51, 68, 85

** **

__Example 5.14__

Find LM,
MN, LN and also the area of ∆ *LON*_{.}

_{}

*Solution:*

From ∆*LMO*, by Pythagoras theorem,

*LM*^{2}* *=* OL*^{2}* *−* OM*^{2}

⇒* LM*^{2}* *=13^{2}* *−12^{2}* *=169* *−144* *=* *25* *=* *5^{2}

∴* LM *=* *5* units*

From ∆*NMO*, by Pythagoras theorem,

*MN*^{2}* *=* ON*^{2}* *−* OM*^{2}

= 15^{2} −12^{2} = 225 −144 = 81= 9^{2}

∴* MN *=* *9* units*

Hence, *LN* = *LM*
+
*MN* = 5 +
9 =
14 *units*

Area of ∆*LON *=* *1/2* *×* base *×* height*

= 1/2 × *LN *× *OM*

= 1/2 × 14 × 12 = 84 *square units.*

** **

__Example 5.15__

∆ABC is equilateral
and CD of the right triangle BCD is 8 *cm*.

Find the
side of the equilateral ∆ABC and also BD.

*Solution:*

As ∆ABC is
equilateral from the figure, AB=BC=AC= (*x*−2)
*cm*.

From ∆*BCD*, by Pythagoras theorem

*BD*^{2}* *=* BC*^{2}* *+* CD*^{2}

⇒
( *x* + 2)^{2} =
( *x* − 2)^{2} +
8^{2}

* x*^{2} + 4*x* + 4 = *x*^{2} − 4*x* + 4 + 8^{2}

⇒
8 *x* = 8^{2}

⇒* x *=* *8*cm*

The side
of the equilateral ∆ *ABC* =
6 *cm* and BD =10 *cm*.

Tags : Geometry | Chapter 5 | 8th Maths , 8th Maths : Chapter 5 : Geometry

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