Question

the legs of a right Triangle are in ratio 3 : 4 and its area is is 101 4 CM square. Find its hypotenuse​

Answer 1

Correct Question:

The length of two short sides of a right Triangle is in ratio 3:4 and its area is 1014 CM square. Find its hypotenuse.

$\star$ Solution:-

Given:

• The ratio of two short sides of the right triangle is 3:4
• The area of the right-angled triangle is 1014 cm sq.

Let, the side i.e Base be 3x cm

And Perpendicular (height) be 4x cm

• Area of the triangle = 1014 cm sq.

$\implies \frac{1}{2}\times base \times height = 1014$

$\implies \frac{1}{2}\times 3x \times 4x = 1014$

$\implies 3x \times 2x = 1014$

$\implies 6x^2 = 1014$

$\implies x^2 = 1014 \div 6$

$\implies x^2 = 169$

$\implies x = \sqrt{169}$

$\implies \boxed{x = 13}$

∴ x = 13

So, Base = 3x = 39cm

And Perpendicular(height) = 4x = 52cm

$\bullet$ So, by using Pythagoras theorem,

$\boxed{\rm{\red{(Hypotenuse)^{2}= (base)^{2}+(perpendicular)^{2}}}}$

⇒ Hypotenuse² = (39)² + (52)²

⇒ Hypotenuse² = 1521 + 2704

⇒ Hypotenuse² = 4225

⇒ Hypotenuse = $\sqrt{4225}$

⇒ Hypotenuse = 65cm

Hence,

The Hypotenuse of the right-angled triangle is 65cm.

Answer 2

Figure :- Refers to the attachment

$\mathtt{\huge{\underline{\red{Question\:?}}}}$

✴ The legs of a right triangle are in ratio 3 : 4 and its area is is 1014 cm². Find its hypotenuse.

$\mathtt{\huge{\underline{\green{Answer:-}}}}$

➡ The hypotenuse of Triangle is 65 cm.

$\bigstar{\mathtt{\huge{\underline{\pink{Solution:-}}}}}$

Given :-

• The legs of a right triangle are in ratio 3 : 4.

• The area of triangle is 1014 cm².

To Find :-

• The hypotenuse of Triangle.

Calculation :-

Let the leg of the triangle be x

So it's sides 3:4 are equal to 3x and 4x .

According to the question,

We know the, Area of right angle triangle = $\</strong><strong>d</strong><strong>frac{</strong><strong>1</strong><strong>}{</strong><strong>2</strong><strong>}$×base×hieght.

➝ 1014 = $\dfrac{1}{2}$ × 3x × 4x

➝ 1014 × 2 = 3x × 4x

➝ 2028 = 12x²

➝ $\dfrac{2028}{12}$ = x²

➝ x² = $\cancel{\dfrac{2028}{12}}$

➝ x² = 169

➝ x = $\sqrt{169}$

➩ x = 13 cm

Using x finding the sides of triangle,

• 3x = 3×13 = 39

• 4x = 4×13 = 52

Here , We have a right triangle,applying Pythagoras theorem.

✒ Pythagoras theorem states the sum of the square of two sides namely, perpendicular & base is equals to the square of its hypotenuse.

$\mathcal{\orange{Hypotenuse²\:=\:Perpendicular²\:+\:Base²}}$

➤ Hypotenuse ² = P² + B²

➠ Hypotaneuse² = 39² + 52²

➠ Hypotaneuse² = (39×39) + (52×52)

➠ Hypotaneuse² = 1521 + 2704

➠Hypotaneuse = $\sqrt{4225}$

➙ Hypotaneuse = 65 cm.

⍈ Hence, The hypotenuse of the triangle is 65 cm.

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