Question

find the area of a triangle with vertices A(1,1),B(1,-4)&C(5,-2)​

Answer 1

$\sf\huge\pink{\underbrace{ Solution : }}$

$\tt\large\blue{\underline{\bigstar{ Given\:that: }}}$

◼ Vertices of a triangle are :

• A(1,1)
• B(1,-4)
• C(5,-2)

$\tt\large\green{\underline{\bigstar\:{To\:find: }}}$

◼ Area of the Triangle.

$\tt\large\orange{\underline{\bigstar\:{Formula : }}}$

By using Area of the triangle formula :

$\boxed{\rm{\red{ \triangle = \cfrac{1}{2} | x_{1}(y_{2} - y_{3}) + x_{2}(y_{3} - y_{1})+x_{3}(y_{1}-y_{2})| }}}$

$\tt\large\purple{\underline{\bigstar\:{Let : }}}$

• x1 = 1 ; y1 = 1
• x2 = 1 ; y2 = - 4
• x3 = 5 ; y3 = -2

$\sf \implies \cfrac{1}{2} | 1(-4-(-2)) + 1(-2-1) + 5(1-(-4)) |$

$\sf \implies \cfrac{1}{2} | 1(-4+2) + 1(-3) + 5(1+4) |$

$\sf \implies \cfrac{1}{2} | 1(-2) -3 + 5(5) |$

$\sf \implies \cfrac{1}{2} | -2 -3 + 25 |$

$\sf \implies \cfrac{1}{2} | 20 |$

$\sf \implies \cancel{\cfrac{20}{2}}= 10$

$\underline{\boxed{\rm{\purple{\therefore Area\:of\:Triangle =10\:sq.units.}}}}\:\orange{\bigstar}$

Answer 2

$\tt \huge{ \underline{ \underline{ \red{Question : - }}}}$

find the area of a triangle with vertices A(1,1),B(1,-4)&C(5,-2)

$\tt \huge{ \underline{ \underline{ \red{Answer : - }}}}$

we know that,

${\star {\boxed{ \bf{ \blue{area \: \: of \: \triangle \: =1/2[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]}}}}}$

$\sf \large \star \: x_1 = 1 \\ \\ \sf \large \star \: x_2 = 1 \\ \\ \sf \large \star \: x_3 = 5 \\ \\ \sf \large \star \: y_1 = 1 \\ \\ \sf \large \star \: y_2 = - 4 \\ \\ \sf \large \star \: y_3 = - 2$

substitute in Formula:-

$\tt \large : \implies \: \frac{1}{2} [1( - 4) - ( - 2) + 1( - 2 - 1) + 5(1) - ( - 4)] \\ \tt \large : \implies \: \frac{1}{2} [1( - 4 + 2) + 1( - 3) + 5(1 + 4)] \\ \\ \tt \large : \implies \: \frac{1}{2} [1( - 2) - 3 + 5(5)] \\ \\ \tt \large : \implies \: \frac{1}{2} [ - 2 - 3 + 25] \\ \\ \tt \large : \implies \: \frac{1}{2} [ - 5 + 25] \\ \\ \tt \large : \implies \: \frac{1}{2} [20 ] \\ \\ \tt \large : \implies \: 10$

Area of triangle (∆) is 10 sq. units