In the given figure, intersect at O.
(a) Determine y, when x = 60°.
(b) Determine x, when y = 40°
Answers
Answer:
Given: The Difference of two numbers is 14. & the difference of Square of these two numbers is 448.
Need to find: The numbers?
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❍ Let's say, the the numbers be x and y respectively.
Given that,
Difference of these two numbers is 14.
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\begin{gathered}:\implies\sf x - y = 14\qquad\qquad\quad\sf\Bigg\lgroup eq^{n}\;(1)\Bigg\rgroup\\\\\end{gathered}
:⟹x−y=14
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eq
n
(1)
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Also,
Difference of Square of these numbers is 448.
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\begin{gathered}:\implies\sf x^2 - y^2 = 448\\\\\\\end{gathered}
:⟹x
2
−y
2
=448
\begin{gathered}:\implies\sf (x + y) (x - y) = 448\qquad\qquad\quad\sf\Bigg\lgroup \Big(x^2 - y^2\Big) = \Big(x + y \Big) \Big(x - y \Big)\Bigg\rgroup\\\\\\\end{gathered}
:⟹(x+y)(x−y)=448
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(x
2
−y
2
)=(x+y)(x−y)
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\begin{gathered}:\implies\sf x + y \times 14 = 448\\\\\\\end{gathered}
:⟹x+y×14=448
\begin{gathered}:\implies\sf x + y = \cancel\dfrac{448}{14}\\\\\\\end{gathered}
:⟹x+y=
14
448
\begin{gathered}:\implies\sf x + y = 32\qquad\qquad\quad\sf\Bigg\lgroup eq^{n}\;(2)\Bigg\rgroup\\\\\\\end{gathered}
:⟹x+y=32
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eq
n
(2)
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⌑ From eqₙ ( I ) & eqₙ ( II ) :
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\begin{gathered}\longrightarrow\sf x + \cancel{\;y} + x - \cancel{~ y}= 32 + 14\\\\\\\end{gathered}
⟶x+
y
+x−
y
=32+14
\begin{gathered}\longrightarrow\sf 2x = 46\\\\\\\end{gathered}
⟶2x=46
\begin{gathered}\longrightarrow\sf x = \cancel\dfrac{46}{2}\\\\\\\end{gathered}
⟶x=
2
46
\begin{gathered}\longrightarrow{\pmb{\underline{\boxed{\frak{x = 23}}}}}\;\bigstar\\\\\end{gathered}
⟶
x=23
x=23
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✇ Putting value of x in eqₙ ( II ) :
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\begin{gathered}\longrightarrow\sf x + y = 32\\\\\\\end{gathered}
⟶x+y=32
\begin{gathered}\longrightarrow\sf 23 + y = 32\\\\\\\end{gathered}
⟶23+y=32
\begin{gathered}\longrightarrow\sf y = 32 - 23\\\\\\\end{gathered}
⟶y=32−23
\begin{gathered}\longrightarrow{\pmb{\underline{\boxed{\frak{y = 9}}}}}\;\bigstar\\\\\end{gathered}
⟶
y=9
y=9
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\therefore{\underline{\textsf{Hence, the numbers are \textbf{23} and \textbf{9} respectively.}}}∴
Hence, the numbers are 23 and 9 respectively.