Question

the side of a right triangle are in the ratio 3 ratio 4 a rectangle is described on its hypotenuse being the long side of the rectangle the breadth of the rectangle is 4 by 5 of a length find the shorter side of the right tringle if the perimeter of the rectangle is 180cm​

Answer 1

Given the ratio between two sides of a right angle triangle is 3 : 4

Let the two sides be 3x and 4x respectively.

Applying Pythagoras theorem we get,

h² = p² + b²

h² = (4x)² + (3x)²

h² = 16x² + 9x²

h = √(25x²)

h = 5x

ATQ,

Hypotenuse of the triangle = length of the rectangle

➡ Length of the rectangle = 5x

And it's breadth = 5x × 4/5 = 4x

Perimeter of the rectangle = 180cm

➡ 2(l + b) = 180cm

➡ 2(5x + 4x) = 180cm

➡ 9x = 180/2

➡ x = 90/9

➡ x = 10cm

Now, we can clearly see that the shortest side of the right angle triangle is 3x.

Therefore the shortest side of the right angle triangle = 3 × 10 = 30m

Answer 2

$\Huge{\underline{\underline{\red{\sf{Answer :}}}}}$

$\large \tt \: given\begin{cases} \sf{Base \: and \: Height \: ratio \: = \: 3:4 } \\ \sf{Perimeter \: of \: Rectangle \: = 180 \: cm} \\ \sf{Hypotenuse \: is \: length \: of \: rectangle } \end{cases}$

Solution :

Let the sides of triangle be 3x and 4x

We have to find Hypotenuse,

Use Pythagoras theorem

$\Large \longrightarrow {\sf{(H)^2 \: = \: (B)^2 \: + \: (P) ^2}}$

Where,

H is hypotenuse

B is Base

P is Perpendicular

Put Values

⇒ H² = (3x)² + (4x)²

⇒H² = 9x² + 16x²

⇒H² = 25x²

⇒H = √25x²

⇒H = 5x

$\Large {\boxed{\sf{Hypotenuse \: = \: 5x}}}$

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Now come to triangle.

Hypotenuse = 5x = Length of triangle

Breadth = 5x * 4:5 = 4x

Now use formula for perimeter of rectangle.

$\Large {\boxed{\sf{Perimeter \: = \: 2(l \: + \: b}}}$

Put Values

⇒ 180 = 2(5x + 4x)

⇒180 = 2(9x)

⇒180 = 18x

⇒x = 180/18

⇒x = 10

Shortest side = 3x = 3(10) ⇒30 cm

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