Question

Find the area of the triangle formed by the points by using the heron's formula (2,3),(-1,3)and(2,-1)

Answer 1

GiVen:

★ Points :

• A(2,-3)

• B(-1,3)

• C(2,-1)

AnSwer:

$\\ \sf \implies \: AB = \sqrt{ {( - 1 - 2)}^{2} + {(3 - 3)}^{2} } \\ \sf \implies \:AB = \sqrt{ { (- 3)}^{2} + {(0)}^{2} } \\ \sf \implies \:AB = \sqrt{9} \\ \implies { \underline{ \boxed{ \sf{ \red{\:AB = 3 \: units }}}}} \: \bigstar\\ \\ \\ \sf \implies \:BC = \sqrt{ {( -2 - ( - 1))}^{2} + {( - 1- 3)}^{2} } \\ \sf \implies \:BC = \sqrt{ {(2 + 1)}^{2} + {( - 4)}^{2} } \\ \sf \implies \:BC = \sqrt{ {(3)}^{2} + 16 } \\ \sf \implies \:BC = \sqrt{9 + 16} \\ \sf \implies \:BC = \sqrt{25} \\ \implies { \underline{ \boxed{ \sf{ \purple{\:BC =5 \: units}}}}} \: \bigstar \\ \\ \\ \sf \implies \:AC =\sqrt{ {( 2 - 2)}^{2} + {( - 1 - 3)}^{2} } \\ \sf \implies \:AC = \sqrt{ {(0)}^{2} + {( - 4)}^{2} } \\ \sf \implies \:AC = \sqrt{16} \\ \implies { \underline{ \boxed{ \sf{ \pink{\:AC =4 \: units}}}}} \: \bigstar$

Now,

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Let us firstly find the value of S.

As we know that,

$\longmapsto \: \: {\boxed{ \boxed{\sf { \purple{\: S = \dfrac{a + b + c}{2} }}}}}$

$\sf\longmapsto \: \:s = \dfrac{5 + 4 + 3}{2} \\ \\ \sf\longmapsto \: \:s = \dfrac{12}{2} \\ \\ \sf\longmapsto { \sf{\: \:s = 6 \: units}}$

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Now, According to the Herons formula,

$\sf \implies A={\sqrt {s(s-a)(s-b)(s-c)}} \\ \\ \sf \implies \: \sqrt{6(6 - 5)(6 - 4)(6 - 3)} \\ \\ \sf \implies \: \sqrt{6 \times 1 \times 2 \times 3} \\ \\ \sf \implies \: \sqrt{36 \: } \\ \\ \implies { \underline{ \boxed{ \sf{ \blue{\: 6 \: units}}}}} \: \bigstar$

Therefore, area of the triangle formed is 6 units.

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