Question

Find the area of triangle whose vertices are (1, –1), (–4, 6) and , (–3, –5) using herone's formulae

Answer 1

Given

• A ( 1 , - 1 )
• B ( - 4 , 6 )
• C ( - 3 , - 5 )

To Find :

• Area of the triangle

Solution :

$\sf Area_{triangle} = \dfrac{1}{2} \bigg[x_1(y_2 -y_2) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \bigg] \\ \\ \sf x_1=1\: \: \:\: \:\:\:\:x_2=-4\: \:\:\:\: \: \:\:\:x_3=-3\\ \\\sf y_1=-1\: \:\:\:\: \: \:\:\: y_2=6\: \:\:\:\: \: \:\:\:y_3=-5 \\ \\\sf \implies \dfrac{1}{2} \bigg[1(6 - ( - 5)) +( - 4)( - 5 - ( - 1)) + ( - 3)( - 1 - 6) \bigg] \\ \\ \sf \implies \dfrac{1}{2} \bigg[1(6 + 5) - 4( - 5 + 1) - 3( - 1 - 6) \bigg] \\ \\ \sf \implies \dfrac{1}{2} \bigg[1(11) - 4( - 4) - 3( - 7) \bigg] \\ \\ \sf \implies \dfrac{1}{2} \bigg[11 + 16 + 21 \bigg] \\ \\ \sf \implies \dfrac{1}{2} \times 48 \\ \\ \sf \implies 24$

$\large \underline{ \bf Area \: of \: triangle = 24 \: {cm}^{2} }$

Answer 2

Step-by-step explanation:

We Have :-

$a(1,- 1) \\ \\ \\ b( - 4,6) \\ \\ \\ c( - 3, - 5)$

To Find :-

$Area \: of \: triangle \: using \: herons \: formula$

Formulas Used :-

$Area \: of \: triangle \: = \: \dfrac{1}{2} (x_{1} (y_{2} - y_{3}) + x_{2}(y_{3} - y_{1}) + x_{3}(y_{1} - y_{2}))$

Solution :-

$area \: = \: \dfrac{1}{2} (x_{1} (y_{2} - y_{3}) + x_{2}(y_{3} - y_{1}) + x_{3}(y_{1} - y_{2})) \\ \\ \\ putting \: the \: values \\ \\ \\ area = \dfrac{1}{2} (1 (6 + 5) - 4( - 5 + 1) - 3( - 1- 6)) \\ \\ \\ area = \dfrac{1}{2} (1 (11) - 4( - 4) - 3( - 7)) \\ \\ \\ area = \dfrac{1}{2} (11 + 16 + 21) \\ \\ \\ area = \dfrac{1}{2} (48) \\ \\ \\ area = 24 \: sq \: units$