Question

prove converse of Pythagoras theorem and write statement.​

Answer 1

Answer:

PYTHAGORAS THEOREM:

STATEMENT:

In a right angle triangle,the square on the hypotenuse

is equal to the sum of the squares on the other two sides.

Proof:

given: In triangle ABC ,angle A=90 degree.

to prove: AB^2+AC^2=BC^2

construction: Draw AD perpendicular to BC

Step-by-step explanation:

Compare triangle ABC and triangle DBA.

Angle B is common

Angle BAC= Angle BDA=90 degree

THEREFORE,triangle ABC is similar to triangle

DBA.

BY AA SIMILARITY,

AB/BD=BC/AB

AB^2=BC*BD............(1)

Compare triangle ABC and DAC

Angle C is common

Angle BAC=Angle ADC=90 degree

THEREFORE,triangle ABC is similar to triangle DAC.

BY AA SIMILARITY,

BC/AC=AC/DC

AC^2=BC*DC...............(2)

Adding (1)&(2)we get

AB^2+AC^2=BC*BD+BC*DC

=BC(BD+DC)=BC*BC

AB^2+AC^2=BC^2

HENCE THE THEOREM IS PROVED.

Answer 2

To prove :-

• ∆ABC is right angle triangle, right angled at angle B.

Step-by-step explanation :

Given :-

• AB² + BC² = AC²

Construction :-

• Construct a right angle triangle PQR so that PQ = AB, QR = BC and angle Q = 90°.

$\purple{\underline{\huge \sf Prove :}}$

Statement : In a triangle, if the square of one side is equal to sum of squares of the other two sides, then the angle opposite to side is right angle.

Concept : For showing ∠B = 90°. First we need to congruent ∆ABC and ∆PQR.

∆PQR is right angle triangle.

So, By Pythagoras theorem [Hypotenuse² = Perpendicular² + Base²] :

$\longrightarrow$ (PR)² = (PQ)² + (QR)²

• By construction PQ = AB and QR = BC.

$\longrightarrow$ (PR)² = (AB)² + (BC)²

• It is given (AB)² + (BC)² = (AC)²

$\longrightarrow$ (PR)² = (AC)²

• Square root both L.H.S and R.H.S

$\longrightarrow$ √(PR)² = √(AC)²

$\longrightarrow$ PR = AC --------(i)

Now, In ∆ABC and ∆PQR :

AB = PQ [By construction]

BC = QR [By construction]

AC = PQ [By equation (i)]

By, SSS congruency,

∆ABC ≅ ∆PQR

By CPCT,

• ∠Q = ∠B [By construction ∠Q is 90°]

So,

• ∠Q = ∠B = 90°

Therefore,

∠B is 90° and,

∆ABC is a right angle triangle.

Hence, Proved!!.