Question

The hypotenuse of a right angle triangle is 13 cm if its base is 7 cm more than its altitude find the base and altitude

Answer 1

Solution :-

Hypotenuse of right angled triangle = 13 cm

Let the altitude of the right angled triangle be x cm

Base is 7 cm more than its altitude = (x + 7) cm

By Pythagoras theorem

⇒ (Base)² + (Altitude)² = (Hypotenuse)²

⇒ (x + 7)² + x² = 13²

⇒ x² + 7² + 2(x)(7) + x² = 169

⇒ 2x² + 49 + 2(7x) - 169 = 0

⇒ 2x² + 2(7x) - 120 = 0

⇒ 2(x² + 7x - 60) = 0

⇒ x² + 7x - 60 = 0

Splitting the middle term

⇒ x² + 12x - 5x - 60 = 0

⇒ x(x + 12) - 5(x + 12) = 0

⇒ (x - 5)(x + 12) = 0

⇒ x - 5 = 0 or x + 12 = 0

⇒ x = 5 or x = - 12

x ≠ - 12 because lengths cannot be negative

⇒ x = 5

Altitude = x = 5 cm

Base = (x + 7) = 5 + 7 = 12 cm

Therefore 12 cm and 5 cm are measurements of base and altitude respectively.

Answer 2

Answer:

$\bold\green{Base}$ = $\huge{\boxed{\mathfrak\green{12\:cm}}}$

$\bold\green{Altitude}$ = $\huge{\boxed{\mathfrak\purple{5\:cm}}}$

Step-by-step explanation:

Given:-

• Hypotenuse = 13 cm
• Base = Altitude + 7

• Base = ?
• Altitude = ?

Let altitude be x cm. So base = ( x + 7) cm

$\bigstar{\underline{\bold{According\:to\:Pythagoras\:theorem:-}}}$

$\Rightarrow\sf {Hypotenuse}^{2} = {Base}^{2} + {Altitude}^{2} \\\\\Rightarrow\sf {13}^{2} = {(x+ 7)}^{2} + {x}^{2} \\\\\Rightarrow\sf 169 = {x}^{2} + 2x \times 7 + {7}^{2} + {x}^{2} \\\\\Rightarrow\sf 169 = 2{x}^{2} + 14x + 49 \\\\\Rightarrow\sf 2{x}^{2} + 14x + 49 - 169 = 0 \\\\\Rightarrow\sf 2{x}^{2} + 14x - 120 = 0 \\\\\Rightarrow\sf 2({x}^{2} + 7x - 60 ) = 0 \\\\\Rightarrow\sf {x}^{2} + 7x - 60 = 0 \\\\\Rightarrow\sf {x}^{2} + 12x - 5x - 60 = 0 \\\\\Rightarrow\sf x(x + 12) - 5(x + 12) = 0 \\\\\Rightarrow\sf (x + 12) (x - 5) = 0 \\\\\Rightarrow\sf (x + 12) = 0 \: \: or, \: \: (x - 5) = 0 \\\\\Rightarrow\sf x = - 12 \: \: \: or, \: \: \: x = 5 \\\\\Rightarrow\sf x \neq -12 \: \: [\because Length \: can't\: be\: negative]$

$\therefore{\boxed{\bold\red{x = 5 \: cm}}}$

$\rule{100}{2}$

● Altitude = x = 5 cm

● Base = (x + 7)= (5 + 7) = 12 cm

$\therefore\bold{Base\:and\:altitude\:are\:respectively\:12\:cm\: \& \: 5 \: cm }$