Question

The perimeter of a right angled triangle is60 cm and its hypotenuse is 25cm then find the remaining twosides

Answer 1

Answer :-

Other two sides are 20 cm and 15 cm.

Explanation :-

Hypotenuse of the right triangle (Hyp) = 25 cm

Let the height of the triangle be 'h' cm

Let the base of the triangle be 'b' cm

Given

Perimeter of the right triangle = 60 cm

⇒ Sum of all sides = 60

⇒ Hyp + b + h = 60

⇒ 25 + b + h = 60

⇒ b + h = 60 - 25

⇒ b + h = 35

⇒ b = 35 - h

In a Right triangle

b² + h² = Hyp²

⇒ (35 - h)² + h² = 25²

[ ∵ b = 35 - h ]

⇒ 35² + h² - 2(35)(h) + h² = 625

[ ∵ (x - y)² = x² - 2xy + y²]

⇒ 1225 + h² - 70h + h² = 625

⇒ 2h² - 70h + 1225 - 625 = 0

⇒ 2h² - 70h + 600 = 0

⇒ 2(h² - 35h + 300) = 0

⇒ h² + 35h + 300 = 0/2

⇒ h² - 35h + 300 = 0

Splitting the middle term

⇒ h² - 20h - 15h + 300 = 0

⇒ h(h - 20) - 15(h - 20)

⇒ (h - 20)(h - 15) = 0

⇒ h - 20 = 0 or h - 15 = 0

⇒ h = 20 or h = 15

i) If h = 20 cm

b = 35 - h = 35 - 20 = 15 cm

ii) If h = 15

b = 35 - h = 35 - 15 = 20 cm

∴ the other two sides are 20 cm and 15 cm.

Answer 2

$\mathfrak{\large{\underline{\underline{Answer:}}}}$

The other two sides of the triangle are 15 cm and 20 cm.

$\mathfrak{\large{\underline{\underline{Step-by-step \:explanation}}}}$

$\textsf{\underline{\underline{Given : }}}$

Perimeter of the right angled triangle = 60 cm

Its Hypotenuse = 25 cm

$\textsf{\underline{\underline{To find :}}}$

Measure of remaining two sides

$\textsf{\underline{\underline{Solution :}}}$

$\textbf{\small{\underline{Let the - }}}$

• Base be x
• Height be y

Perimeter of the triangle = Sum of all sides

★ $\green{\boxed{\green{\boxed{\red{\sf{25+x+y=60}}}}}}$

$\sf{\implies} \: 25 + x + y = 60 \\ \sf{\implies} \:x + y = 60 - 25 \\ \sf{\implies} \:x + y = 35 \\ \sf{\implies} \:x = 35 - y \: .....(Eq\:1)$

$\rule{300}{1.5}$

$\textbf{\small{\underline{As we know that - }}}$

★ $\pink{\boxed{\purple{\boxed{\orange{\mathsf{{x}^{2} + {y}^{2} = {Hypotenuse}^{2}}}}}}}$

$\sf{\implies} \: {x}^{2} + {y}^{2} = {Hypotenuse}^{2} \\ \sf{Place \: the \: value \: of \: x \: from \: (Eq \: 1)} \\ \sf{\implies} \: {(35 - y)}^{2} + {y}^{2} = {25}^{2} \\ \sf{\implies} \: {35}^{2} + {y}^{2} - 2(35)(y) + {y}^{2} = 625 \\ \sf{\implies} \: Identity \rightarrow \pink{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2}} \\ \sf{\implies} \:1225 + {y}^{2} - 70y + {y}^{2} = 625 \\ \sf{\implies} \: {2y}^{2} - 70y+ 1225 - 625 = 0 \\ \sf{\implies} \:2 {y}^{2} - 70y + 600 = 0 \\ \sf{\implies} \:2( {y}^{2} - 35h + 300) = 0 \\ \sf{\implies} \: {y}^{2} + 35y + 300 = \frac{0}{2} \\ \bf{\implies} \: {y}^{2} - 35y + 300 = 0</p><p>$

$\rule{300}{1.5}$

$\textbf{\small{\underline{By Middle Term Splitting - }}}$

$\sf{\implies} \: {y}^{2} - 20y - 15y + 300 = 0 \\ \sf{\implies} \: y(y - 20) - 15(y - 20) \\ \sf{\implies} \: (y - 20)(y - 15) = 0 \\ \sf{\implies} \: \red{y - 20 = 0} \: or \: \red{y - 15 = 0} \\ \bf{\implies} \: y = 15 \: or \: 20$

$\rule{300}{1.5}$

$\textbf{\small{\underline{If the value of y is 20 cm }}}$

$\sf{\implies} \: x = 35 - y \\\sf{\implies} \: 35 - 20 \\\bf{\implies} \: 15 \:m$

$\rule{300}{1.5}$

$\textbf{\small{\underline{If the value of y is 15 cm }}}$

$\sf{\implies} \: x= 35 - y \\ \sf{\implies} \: 35 -15 \\ \bf{\implies} \: 20 \: cm$

$\therefore$ The other two sides of the triangle are 15 cm and 20 cm.