Question

In the figure the triangle is right triangle at *Y* and *N* is the foot of the perpendicular from *Y* to XZ. Given that XY=6 cm and XZ=10 cm. What is the length of XN

Answer 1

Answer:

$\underline{ \underline{ \red{ \large {\tt{✧ G \: I \: V \: E \: N}}} }}:$

• XY = 6 cm , XZ = 10 cm

$\underline{ \underline{ \purple{ \large{ \tt{✧ T \: O \: \: F\: I\: N\: D}}}}} :$

• Length of XN

$\underline{ \underline{ \pink{\large {\tt{✧ S \: O \: L \: U\: T\: I \: O\: N}}}} }:$

• Let the length of XN be ' x ' & that of NZ be ' 10 - x '

In rt. XYZ :

• Hypotenuse (h ) = 10 , Perpendicular ( p ) = YZ & Base ( b ) = 6

Using Pythagoras theorem:

$\large{ \sf{p = \sqrt{ {h}^{2} - {b}^{2} } }}$

Plug the values & then simplify!

⇢ $\large{ \sf{YZ = \sqrt{( {10)}^{2} - {(6)}^{2} } }}$

⇢ $\large{ \sf{YZ = \sqrt{100 - 36}}}$

⇢ $\large{ \sf{YZ = \sqrt{64}}}$

⇢ $\large{ \sf{YZ = 8} \: cm} \:$

In rt. YNZ ,

• Hypotenuse = 8 cm , Perpendicular = YN , base = 10 - x

Using Pythagoras theorem :

⇾ $\large{ \sf{ {p}^{2} = {h}^{2} - {b}^{2} }}$

⇾ $\large{ \sf{ {YN}^{2} = {8}^{2} - {(10 - x)}}}^{2}$

⇾ $\large{ \sf{ {YN}^{2} = 64 - \{ {(10)}^{2} - 2 \times 10 \times x + {(x)}^{2} \} }}$

⇾ $\large{ \sf{ {YN}^{2} = 64 - \{100 - 20x + {x}^{2} \}}}$

⇾ $\large{ \sf{ {YN}^{2} = 64 - 100 + 20x - {x}^{2} }}$

⇾ $\large{ \sf{ {YN}^{2} = 20x - {x}^{2} - 36}} \longrightarrow \: eq (i)$

In rt. XYN ,

• Hypotenuse = 6 , Perpendicular = YN , base = x

Using Pythagoras theorem :

$\large{ \sf{ {p}^{2} = {h}^{2} - {b}^{2} }}$

⇾ $\large{ \sf{ {YN}^{2} = {(6)}^{2} - {(x)}}}^{2}$

⇾ $\large{ \sf{ {YN}^{2} = 36 - {x}^{2} \longrightarrow \: eq(ii) }}$

Now , From Equation ( i ) & Equation ( ii )

⤑ $\large{ \sf{20x - {x}^{2} - 36 = 36 - {x}^{2} }}$

⤑ $\large{ \sf{20x - \cancel{ {x}^{2} } - 36 = 36 - \cancel{ {x}^{2} }}}$

⤑ $\large{ \sf{20x = 36 + 36}}$

⤑ $\large{ \sf{20x = 72}}$

⤑ $\large{ \sf{x = 3.6}}$ cm

Hence, The length of XN is $\large{ \boxed{\red{ \bold{ \tt{3.6 \: cm}}}}}$

[ Correct me if I am wrong ]

♨ Hope I helped! ♡

♪ Have a wonderful day / night ! ☃

☥ $\underbrace{ \overbrace{{ \mathfrak{Carry \: On \: Learning}}}}$ ✎

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Answer 2

Answer:

3.6cm

Step-by-step explanation:

YZ = 10² - 6² = 8

sinX = 8/10 = 53.13º

cos 53.13º = XN/6

XN = 3.6cm