Question

Coordinates of the vertices of a triangle are P (-5, -1), Q (3, -5), R (5, 2). What is the area of triangle PQR?

Answer 1

Answer:-

Given Points are P( - 5 , - 1) , Q (3 , - 5) , R(5 , 2).

We know that,

Area of a triangle with co - ordinates

$\sf (x_1 , y_1) , (x_2 , y_2) \:\&\: (x_3 , y_3)$ is :

$\sf \dfrac{1}{2} \begin{vmatrix} \sf \: x _{1} - x _{2} & \sf \: x _{1} - x _{3} \\ \\ \sf \: y _{1} - y _{2}& \sf \: y _{1} - y _{3} \end{vmatrix} \quad$

Let,

• x1 = - 5

• x2 = 3

• x3 = 5

• y1 = - 1

• y2 = - 5

• y3 = 2

Hence,

$\sf \implies \: Area \: of \: \triangle \: PQR = \dfrac{1}{2} \begin{vmatrix} \sf \: - 5 - 3& \sf - 5 - 5 \\ \\ \sf \: - 1 - ( - 5)& \sf \: - 1 - 2\end {vmatrix} \\ \\ \sf \implies \: Area \: of \: \triangle \: PQR = \dfrac{1}{2} \begin{vmatrix} \sf - 8& \sf - 10 \\ \sf 4& \sf- 3 \end{vmatrix} \\ \\ \sf \implies \: Area \: of \: \triangle \: PQR = \dfrac{ 1}{2} | ( - 8)( - 3) -( 4)( - 10)| \\ \\ \sf \implies \: Area \: of \: \triangle \: PQR = \dfrac{1}{2} |24 + 40| \\ \\ \sf \implies \: Area \: of \: \triangle \: PQR = \frac{1}{2} \times 64 \\ \\ \sf \implies \large \red{ Area \: of \: \triangle \: PQR = 32 \:unit ^{2} }$

Therefore, the area of given triangle PQR is 32 unit².

Answer 2

Given :-

• Coordinates of P = ( - 5 , - 1 )

• Coordinates of P = ( 3 , - 5 )

• Coordinates of R = ( 5 , 2 )

To Find :-

• Area of the triangle made by adjoining of given vertices

Solution :-

$\implies \boxed{ \bf \red {Area = \frac{1}{2} \bigg[x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \bigg]}}$

Here

$\tt x_1 = - 5 \: \: \: \: \: \: \: \: \: \: x_2 = 3 \: \: \: \: \: \: \: \: \: \: \: \: x_3 = 5 \\ \\ \tt y_1 = -1 \: \: \: \: \: \: \: \: \: \: y_2 = - 5 \: \: \: \: \: \: \: \: \: \: \: \: y_3 = 2$

Substitute values in formula

$\implies \sf Area = \dfrac{1}{2} \bigg[ - 5( - 5 - 2) + 3(2 + 1) + 5( - 1 + 5) \bigg]\\ \\\implies \sf Area = \dfrac{1}{2} \bigg[- 5( -7) + 3(3) + 5(4) \bigg]\\ \\\implies \sf Area = \dfrac{1}{2} \bigg[35 + 9 + 20\bigg] \\ \\\implies \sf Area = \frac{1}{2} \times64\\ \\ \implies \underline{\boxed{\sf \blue{ Area = 32 \: cm^2}}}$