Question

8.In a right angle triangle the base is 6cm and sum of hypotenuse and the altitude is 18cm .Find the sides of the triangle and then find the perimeter of the triangle​

Answer 1

Given:

• Right angled triangle
• base= 6cm
• Sum of hypotenuse and altitude = 18cm

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Need to Find:

• The perimeter =?
• The sides=?

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Solution:

Perimeter of a triangle= Sum of three sides of triangle

→ Perimeter of right angled triangle = Base+ Height+ Hypotenuse

→ Perimeter= 6cm+ 18cm

→ $\red{\bold{Perimeter = 24cm}}$

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Given:

Perpendicular + Hypotenuse = 18cm

Thus:

P+H= 18cm

H= 18-P cm----i

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By Pythagoras theorem we know,

${hypotenuse}^{2}={perpendicular}^{2}+{Base}^{2}$

⟹${H}^{2}={P}^{2}+{6cm}^{2}$

⟹${18-P}^{2}={P}^{2}+36$

⟹${18}^{2}-2×18×P+{P}^{2}={P}^{2}+36$

⟹${18}^{2}-36P= 36$

⟹324- 36P= 36

⟹36P= 324-36

⟹36P= 288

⟹P= 288÷ 36cm

⟹P= 8cm

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Then H:

= 18- P

= 18-8cm

=10cm

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$\therefore$ The three sides are:

• $\red{Perpendicular= 8cm}$
• $\red{Hypotenuse=10cm}$
• $\red{Base= 6cm}$

Answer 2

To Find :-

• The three sides of the triangle

• The perimeter of the triangle

Given :-

• Base of the triangle = 6 cm

• Sum of hypotenuse and the height = 18 cm

We know :-

Pythagoras Theorem :-

$\boxed{\underline{\bf{h^{2} = p^{2} + b^{2}}}}$

Where :-

• h = Hypotenuse of the triangle

• p = Height of the triangle

• b = Base of the triangle

Perimeter of a Triangle :-

$\boxed{\underline{\bf{P = a_{1} + a_{2} + a_{3}}}}$

Where :-

• a = Side of the t

Concept :-

Let the height be p cm and the hypotenuse be h cm.

According to the Question, the sum of height and hypotenuse of the triangle is 18 cm.i.e,

$:\implies \underline{\underline{\bf{h + p = 18}}}$

From this information , we can find the value of hypotenuse or height in terms of height and hypotenuse respectively.

$:\implies \underline{\underline{\bf{h = 18 - p}}}$

Hence, the value of hypotenuse in terms of height is (18 - p).

Then, by using the Pythagoras theorem , we can find the value of p .i.e, height and putting the value of p in the value of h (in terns of p) to find the hypotenuse .

After finding the sides , we can use the formula and can find the perimeter of the triangle.

Solution :-

Sides of the triangle :-

The Height of the triangle —

Given :-

• Hypotenuse = (18 - p) cm

• Height = p cm

• Base = 6 cm

Using the formula and substituting the values in it, we get :-

$:\implies \bf{h^{2} = p^{2} + b^{2}} \\ \\ \\ :\implies \bf{(18 - p)^{2} = p^{2} + 6^{2}} \\ \\ \\ :\implies \bf{18^{2} - 2 \times 18 \times p + p^{2} = p^{2} + 36}$

$\boxed{\begin{minipage}{8 cm} Here, (18 - p)^2 is written as \\ \bf{18^{2} - 2 \times 18 \times p + p^{2}} \\ by using the identity :- \\ \\ \bf{\underline{(a + b)^{2} = a^{2} + 2ab + b^{2}}} \end{minipage}}$

$:\implies \bf{18^{2} - 36p + p^{2} = p^{2} + 36} \\ \\ \\ :\implies \bf{18^{2} - 36p + p^{2} - p^{2} = 36} \\ \\ \\ :\implies \bf{18^{2} - 36p = 36} \\ \\ \\ :\implies \bf{324 - 36p = 36} \\ \\ \\ :\implies \bf{ - 36p = 36 - 324} \\ \\ \\ :\implies \bf{ - 36p = - 288} \\ \\ \\ :\implies \bf{ \not{-} 36p = \not{-} 288} \\ \\ \\ :\implies \bf{p = \dfrac{288}{36}} \\ \\ \\ :\implies \bf{p = 8} \\ \\ \\ \therefore \purple{\bf{p = 8 cm}}$

Hence, the height of the triangle is 8 cm.

Hypotenuse :-

Given :-

• h = 18 - p

• p = 8 cm

Putting the value of p in the hypotenuse , we get :-

$:\implies \bf{h = 18 - p} \\ \\ :\implies \bf{h = 18 - 8} \\ \\ :\implies \bf{h = 10} \\ \\ \therefore \purple{\bf{h = 10 cm}}$

Hence, the Hypotenuse of the triangle is 10 cm.

Hence, the sides of the triangle are 10 cm and 8 cm.

Perimeter of the triangle :-

Given :-

• $a_{1}$ = 10 cm

• $a_{2}$ = 8 cm

• $a_{3}$ = 6 cm

Using the formula and substituting the values in it we get :-

$:\implies \bf{P = a_{1} + a_{2} + a_{3}} \\ \\ \\ :\implies \bf{P = 10 + 8 + 6} \\ \\ \\ :\implies \bf{P = 24} \\ \\ \\ \therefore \purple{\bf{P = 24 cm}}$

Hence, the perimeter of the triangle is 24 cm.