Question

Determine whether the triangle, the lengths of whose sides are 2.1 cm, 2.8 cm and 3.5 cm is a right-angled triangle.

Answer 1

Answer:

$\large{\underline{\underline{\sf{Firstly, \; let's \; understand \; the \; concept \; used\; :-}}}}$

Here the concept of Pythagoras Theorem has been used. We know that, according to the Pythagoras Theorem, the square of Hypotenuse is equal to the sum of squares of its base and height. For checking this triangle, we can use this theorem.

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★ Formula Used :-

$\: \large{\boxed{\boxed{\sf{(Hypotenuse)^{2} \; \: \: = \: \: \bf{(Base)^{2} \: \; + \: \; (Height)^{2} }}}}}$

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★ Question :-

Determine whether the triangle, the lengths of whose sides are 2.1 cm, 2.8 cm and 3.5 cm is a right-angled triangle.

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★ Solution :-

Given,

» Sides of the triangle = 2.1 cm, 2.8 cm and 3.5 cm

Here we see that the longest side is 3.5 cm so surely it will be the Hypotenuse. And let the base and height be the other two sides.

So applying Pythagoras Theorem their, we get

$\: \qquad \large{\sf{:\Longrightarrow \: \; \: \: (Hypotenuse)^{2} \; \: \: = \: \: \bf{(Base)^{2} \: \; + \: \; (Height)^{2}}}}$

$\: \qquad \large{\sf{:\Longrightarrow \: \: \; \: (3.25 \; cm)^{2} \: \: = \: \: \bf{(2.1\; cm)^{2} \: \: + \: \: (2.8 \; cm)^{2}}}}$

$\: \qquad \large{\sf{:\Longrightarrow \: \; \: \: 12.25\; cm^{2} \: \: = \: \: \bf{4.41\; cm^{2} \: \: + \: \: 7.84\; cm^{2}}}}$

$\: \qquad \large{\sf{:\Longrightarrow \: \; \: \: 12.25\; cm^{2} \: \: = \: \: \bf{12.25\;cm^{2}}}}$

We see that the Pythagoras Theorem is proved here. So the triangle forms right angled triangle.

Clearly here the condition satisfies, hence the triangle is right angle

• Hence, the triangle of sides 2.1 cm, 2.8 cm and 3.5 cm forms a right angled triangle.

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~ Evaluation from the Figure.

From figure attached,we see :-

• BD = 3.5 cm = Hypotenuse

• CD = 2.1 cm = Base

• BC = 2.8 cm = Height

From this, by applying Pythagoras Theorem, we get,

=> (BD)² = (CD)² + (BC)²

=> (3.5 cm)² = (2.1 cm) + (2.8 cm)

=> 12.25 cm² = 12.25 cm²

Hence this triangle is Right Angled Triangle where <BCD = 90°.

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~ More to know :-

• Acute Angled Triangle is a triangle whose all the angles are acute angles.

• Obtuse Angled Triangle is a triangle whose one angle is greater than 90°.

• Right Angled Triangle is a triangle whose one angle is equal to 90°.

• Scalene Triangle is a triangle is all the sides are of different lengths.

• Isosceles Triangle is a triangle whose two sides are equal in length.

• Equilateral Triangle is a triangle whose all the angles are equal to 60°.

• Converse of Pythagoras Theorem states that if the square of Hypotenuse is equal to the sum of square of Base and Height of the triangle, then the triangle is a Right Angled Triangle.

Answer 2

Given:-

• Sides of a triangle are: 2.1 cm, 2.8 cm and 3.5 cm

To find:-

• Determining whether the triangle with sides 2.1 cm, 2.8 cm and 3.5 cm will form a right-angled triangle.

Knowledge required:-

Converse of Pythagoras theorem

• This theorem states that, If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.

For a triangle with lengths of sides., AB, BC and AC where AB is the longest side. If

AB² = AC² + BC²

Then By the converse of Pythagoras theorem

Triangle ABC would be a right-angled triangle., right-angled at C.

Solution:-

Given three lengths are:

2.1 cm, 2.8 cm and 3.5 cm

Here, the longest side is AB = 3.5 cm

and the other two sides are BC = 2.1 cm and AC = 2.8 cm

So,

→ AB² = ( 3.5 cm )² = 12.25 cm²

and

→ BC² + AC² = ( 2.1 cm )² + ( 2.8 cm )²

→ BC² + AC² = 4.41 cm² + 7.84 cm²

→ BC² + AC² = 12.25 cm²

Now we can clearly see that,

AB² = BC² + AC²

Hence.,

Using the Converse of Pythagoras theorem we can conclude that,

• Triangle formed with the sides 2.1 cm, 2.8 cm and 3.5 cm is a Right-angled triangle.