Question

One side of a right angle triangle measure 126cm and difference in length of its hypotenuse and other side is 42cm. find the measure of its two unknown sides and calculate the area.

Answer 1

Solution :

$\bf{\red{\underline{\bf{Given\::}}}}$

One side of a right angled triangle measure 126 cm and difference in length of it's hypotenuse and other side is 42 cm.

$\bf{\red{\underline{\bf{To\:find\::}}}}$

The measure of it's two unknown sides and the area.

$\bf{\red{\underline{\bf{Explanation\::}}}}$

In right angled Δ ABC :

We have;

• One side (BC) of right angled Δ  = 126 cm
• Other side of right angled Δ = 42 cm

A/q

$\longrightarrow\sf{AC-AB=42...........(1)}$

&

We know that formula of the Pythagoras Theorem :

$\boxed{\bf{(Hypotenuse)^{2} =(Base)^{2} +(Perpendicular)^{2} }}}}$

$\longrightarrow\tt{(AC)^{2} =(AB)^{2} +(BC)^{2} }\\\\\\\longrightarrow\tt{(AC)^{2} -(AB)^{2} =(BC)^{2} }\\\\\\\longrightarrow\tt{(AC+AB)(AC-AB)=(126)^{2} \:\:\:[\therefore \:Using\:a^{2} -b^{2}]}\\ \\\\\longrightarrow\tt{(AC+AB)(42)=15876}\\\\\\\longrightarrow\tt{AC+AB=\cancel{\dfrac{15876}{42} }}\\\\\\\longrightarrow\tt{AC+AB=378}\\\\\\\longrightarrow\tt{AC=378-AB.......................(2)}$

Putting the value of AC in equation (1),we get;

$\longrightarrow\tt{378-AB-AB=42}\\\\\\\longrightarrow\tt{378-2AB=42}\\\\\\\longrightarrow\tt{-2AB=42-378}\\\\\\\longrightarrow\tt{-2AB=-336}\\\\\\\longrightarrow\tt{AB=\cancel{\dfrac{-336}{-2} }}\\\\\\\longrightarrow\tt{\pink{AB=168\:cm}}$

Putting the value of AB in equation (2),we get;

$\longrightarrow\tt{AC=(378-168)cm}\\\\\\\longrightarrow\tt{\pink{AC=210\:cm}}$

Now;

We know that formula of the area of triangle :

$\boxed{\bf{Area=\frac{1}{2} \times base\times height\:\:\:\:(sq.unit)}}}}}$

• Base (BC) = 126 cm
• Height (AB) = 168 cm

$\longrightarrow\sf{\bigg(\dfrac{1}{\cancel{2}} \times\cancel{ 126} \times 168\bigg)cm^{2} }\\\\\\\longrightarrow\sf{\big(63\times 168\big)cm^{2}}\\\\\\\longrightarrow\sf{\pink{10584\:cm^{2} }}$

Thus;

The area of right angled triangle is 10584 cm² .

Answer 2

Solution:

one side of right angle triangle = 126cm

other side of right angle triangle = 42cm

☞ Let AC= 126m

☞AB = a+42

By Pythagoras theoram we have,

→ (AB)²= (BC)²+(AC)²

→(a+42)² = (a)²+(126)²

→a²+84a+1764=a²+15876

→84a+1764=15876

→84a = 15876-1764

→84a = 14112

→a = 14112/84

→a= 168

Now ,

☞BC =a=168m

☞ AC = a+2 = 168+42= 210m

→Area of ∆ABC = 1/2×B×H

◕B = a=168

◕H = 126

→1/2×168×126

→84×126

→10584m²

Verification:

By heron's formula,

here

☞s = (a+126+a+42)/2

☞s= (2a+168)/2

☞s= a+84

☞s= 168+84

☞s= 252

Area of ∆ = √[s(s-a)(s-b)(s-c)]]

☞√[252(252-168)(252-126)(252-210)

☞√[(252)(84)(126)(42)

☞√(126×2)(42×2)(126)(42)

☞√(126)²(2)²(42)²

☞126×42×2

☞10584m²