Question

find the area of triangle formed by points by using herons formula.(1,1),(1,4),(5,1)

Answer 1

Answer:

⋆ DIAGRAM :

$\setlength{\unitlength}{1.3cm}\begin{picture}(6,8)\linethickness{0.5mm}\qbezier(1,.5)(2,1)(4,2)\qbezier(4,2)(2,3)(2,3)\qbezier(2,3)(2,3)(1,0.5)\put(.7, .2){ C }\put(4.05, 1.9){ B }\put(1.8,3.1){ A }\put(2.1,3.1){\sf{(1, 1)}}\put(1,0.2){\sf{(5, 1)}}\put(4.35,1.9){\sf{(1, 4)}}\end{picture}$

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$\bigstar\:\underline{\sf Distance\: Formula:} \\ \boxed{ \sf D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} }$

☯ Using Distance Formula :

$\sf AB=\sqrt{(1-1)^2+(4-1)^2}=3\\\\\sf BC=\sqrt{(5-1)^2+(1-4)^2}=5 \\\\\sf AC=\sqrt{(5-1)^2+(1-1)^2}=4$

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★ Semi Perimeter of Triangle :

$\dashrightarrow\sf\:\:Semi\: Perimeter=\dfrac{Sum\:of\:Sides}{2}\\\\\\\dashrightarrow\sf\:\:s=\dfrac{3+5+4}{2}\\\\\\\dashrightarrow\sf\:\:s=\dfrac{12}{2}\\\\\\\dashrightarrow\sf\:\:s=6$

⠀

★ Area of the Triangle Formed :

$:\implies\sf Area=\sqrt{s(s-a)(s-b)(s-c)}\\\\\\:\implies\sf Area=\sqrt{6(6-3)(6-5)(6-4)}\\\\\\:\implies\sf Area=\sqrt{6 \times 3 \times 1 \times 2}\\\\\\:\implies\sf Area=\sqrt{6 \times 6}\\\\\\:\implies\underline{\boxed{\sf Area=6 \:unit^{2}}}$

⠀

$\therefore\:\underline{\textsf{Area of the Triangle formed is \textbf{6 unit ^\text2 }}}.$

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⠀⠀⠀⠀⠀⠀⠀Shortcut Trick

In case you don't want to follow Heron's Formula. Then you can use Simple Area Formula of Triangle.

We find out the sides of Triangle as 3, 4 and 5. These are Pythagorean Triplet & so this Triangle is Right Angled.

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★ Area of the Triangle Formed :

$\longrightarrow\sf\:\:Area=\dfrac{1}{2} \times Base \times Height\\\\\\\longrightarrow\sf\:\:Area=\dfrac{1}{2} \times 3 \times 4\\\\\\\longrightarrow\sf\:\:Area=3 \times 2\\\\\\\longrightarrow\:\:\underline{\boxed{\sf Area=6\:unit^2}}$

⠀

$\therefore\:\underline{\textsf{Area of the Triangle formed is \textbf{6 unit ^\text2 }}}.$

Answer 2

Given :

• (1,1),(1,4),(5,1)

To find :

• find the area of triangle

Solution :

D = √(x_2 - x_1)² + (y_2 - y_1)²

AB = √(1 - 1)² + (4 - 1)² = 3

BC = √(5 - 1)² + (1 - 4)² = 5

AC = √(5 - 1)² + (1 - 1)² = 4

Area of triangle : - 1/2 x height x base

area of triangle = 1/2 x 4 x 3

area of triangle = 6 unit²

area of triangle formed is 6 unit²