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

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

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Elliana 1 month 2021-08-17T05:38:11+00:00 2 Answers 0 views 0

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    0
    2021-08-17T05:39:58+00:00

    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.

    _____________________________________

    0
    2021-08-17T05:40:10+00:00

    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.

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