Question

28
The hypotenuse of a right triangle is 65cm
and as area is 15/2cm square. Calculate the
lengths of its perpendiculars sides?
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Answer 1

Correct Question :-

• The hypotenuse of a right triangle is 65cm and as area is 750 cm square. Calculate the lengths of its perpendiculars sides?

Given :-

• The hypotenuse of a right triangle is 65cm.
• Area of triangle is 750 cm square.

To Calculate :-

• Lengths of its perpendiculars sides

Identity used :-

$\large \red{\bf \: ⟼ Pythagoras \: Theorem } ✍$

$\begin{gathered}{\boxed{\bf{\pink{(Hypotenuse)^2 = (Base)^2 + (Perpendicular)^2}}}}\end{gathered}$

$\boxed{\bf \:Area \: of \: triangle = \dfrac{1}{2} \times base \times height}$

$\large \red{\bf \: ⟼ Solution :- } ✍$

$\begin{gathered}\bf\red{Let,} \end{gathered}$

$\begin{gathered}\longmapsto\:\:\bf{Height \:is \:y\: cm}. \\ \end{gathered}$

$\begin{gathered}\longmapsto\:\:\bf{Base \:is \:x\:cm}. \end{gathered}$

$\begin{gathered}\bf\green{According\:to\:the\:question,} \\ \end{gathered}$

${\bf \:Area \: of \: triangle = \: 750}$

${\bf \:Area \: of \: triangle = \dfrac{1}{2} \times base \times height}$

$\bf \: ⟼ 750 = \dfrac{1}{2} \times x \times y$

$\bf \: ⟼ xy = 1500 \: ⟼ (1)$

$\begin{gathered}\bf\pink{Now} \\ \end{gathered}$

$\begin{gathered}\bf\red{hypotenuse \: = 65 \: cm} \end{gathered}$

We know,

$\begin{gathered}{\boxed{\bf{\pink{(Hypotenuse)^2 = (Base)^2 + (Perpendicular)^2}}}}\end{gathered}$

$\bf \: ⟼ {x}^{2} + {y}^{2} = {65}^{2}$

On substituting the value of y, from equation (1), we get

$\bf \: ⟼ {x}^{2} + ({\dfrac{1500}{x} })^{2} = 4225$

$\bf \: ⟼ {x}^{2} + \dfrac{22500}{ {x}^{2} } = 4225$

$\bf \: ⟼ {x}^{4 } - 4225 {x}^{2} + 22500 = 0$

$\bf \: ⟼ {x}^{4} - 3600 {x}^{2} - 625 {x}^{2} + 22500 = 0$

$\bf \: ⟼ {x}^{2} ( {x}^{2} - 3600) - 625( {x}^{2} - 3600) = 0$

$\bf \: ⟼ ( {x}^{2} - 3600)( {x}^{2} - 625) = 0$

$\bf \: ⟼ {x}^{2} = 3600 \: or \: {x}^{2} = 625$

$\bf \: ⟼ x = 60 \: or \: x = 25$

$\bf\implies \:y = 25 \: or \: y = 60$

$\bf \:\large \red{Length \: of \: perpendicular \: sides \: are \:} ✍$

$\bf \: ⟼ 60 \: cm \: and \: 25 \: cm$

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