Question

3.
The height of a triangle is 3 cm less than its base. The area of the triangle is 90 cm. Find the
base of the triangle.​

Answer 1

Given : The height of a triangle is 3 cm less than its base & the area of the triangle is 90 cm² .

Need To Find : The Base of triangle ?

⠀⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━⠀

❍ Let's Consider Base of Triangle be x cm .

⠀⠀⠀⠀⠀As , Per the Question, the height of a triangle is 3 cm less than its base .

Therefore,

• Height of Triangle is ( x - 3 ) cm .

As , We know that ,

$\qquad \star \:\:\: \underline {\boxed{ \pmb{\frak{ \pink{ \:\: Area_{(\:Triangle \:)}\:=\: \dfrac{1}{2}\times b \times h \:\:}}}}}$

⠀⠀⠀⠀⠀Here , b is the Base of Triangle , h is the Height of the Triangle & Area of Triangle is 90 cm² .

$\qquad \dashrightarrow \sf \: Area_{(\:Triangle \:)}\:=\: \dfrac{1}{2}\times b \times h \:$

⠀⠀⠀⠀⠀⠀$\underline {\boldsymbol{\star\:Now \: By \: Substituting \: the \: known \: Values \::}}$

$\qquad \dashrightarrow \sf \: Area_{(\:Triangle \:)}\:=\: \dfrac{1}{2}\times b \times h \:$

$\qquad \dashrightarrow \sf \: 90\:=\: \dfrac{1}{2}\times x \times \bigg(\: x - 3\:\bigg) \:$

$\qquad \dashrightarrow \sf \: 90 \: \times 2 \:=\: x \times \bigg(\: x - 3\:\bigg) \:$

$\qquad \dashrightarrow \sf \: 180 \:=\: x \times \bigg(\: x - 3\:\bigg) \:$

$\qquad \dashrightarrow \sf \: 180 \:=\: \bigg(\: x^2 - 3x\:\bigg) \:$

$\qquad \dashrightarrow \sf \: 180 \:=\: \: x^2 - 3x\: \:$

$\qquad \dashrightarrow \sf \: \: \: x^2 - 3x\: - 180 \:=0 \:$

$\qquad \dashrightarrow \sf \: \: \: x^2 +12x\:-\:15x \: - 180 \:=0 \:$

$\qquad \dashrightarrow \sf \: \: \: x( \:x +12\:)\:-\:15\:(\: x \: + 12\:) \:=0 \:$

$\qquad \dashrightarrow \sf \: \: \: ( \:x - 15 \:)\:\:(\: x \: + 12\:) \:=0 \:$

$\qquad \dashrightarrow \sf \: x\: \:=\:15 \:\:or\:\:-12 \:$

• As , We know that , the measurement of base and triangle cannot be in negative ( -ve ) . So , By ignoring negative value . We get ,

$\qquad \dashrightarrow \sf \: x\: \:=\:15 \:\:or\:\:-12 \:$

$\qquad \dashrightarrow \sf \: x\: \:=\:15 \:\: \:$

$\qquad \therefore \:\:\pmb{\underline{\boxed{\purple{\frak{\:\:\:x \:=\: 15 \: cm \:\: }}}} }\:\:\bigstar$

Therefore,

• The Base of Triangle is x = 15 cm .
• The Height of Triangle is ( x - 3 ) = ( 15 - 3 ) = 12 cm

$\qquad \therefore \:\:\underline {\sf \:Hence \:,\:The \:Base \:of \:Triangle \:is \: \bf 12 \: cm \:}.$