Question

(4)...Plot the points A(1,-4), B(-2, 4) and C(4, 1). Name and shade the figure so obtained of joining them in order and also, find its area and perimeter.​

Answer 1

Refer The Attachment ⬆️

||GIVEN||

• A(1,-4) $\longmapsto\;\sf{(x_1,y_1)}$
• B(-2,4) $\longmapsto\;\sf{(x_2,y_2)}$
• C(4,1) ) $\longmapsto\;\sf{(x_3,y_3)}$

||TO FIND||

• Shape Formed by joining A,B & C.
• Area & Perimeter of the figure formed.

||SOLUTION||

Area of ∆ABC

$\frac{1}{2} |1(4 - 1) + ( - 2)(1 - [ - 4]) + 4( - 4 - 4) |\\ \frac{1}{2} |1(3) - 2(1 + 4) + 4( - 8)| \\ \frac{1}{2} |3 - 10 - 32| \\ \frac{1}{2} |3 - 42| \\ \frac{1}{2} |- 39| \\ \frac{1}{2} \times ( 39) \\ 19.5$

Perimeter of ∆ABC

Length of AB:-

$\sqrt{{ {(1 + 2)}^{2} } + {( - 4 - 4)}^{2} } \\ \sqrt{ {(3)}^{2} +{ ( - 8) }^{2} } \\ \sqrt{(9 + 64)} \\ \sqrt{73} \\ 8.544$

Length of BC:-

$\sqrt{ {( 4 + 2)}^{2} + {(1 - 4)}^{2} } \\ \sqrt{ {(6)}^{2} + {( - 3)}^{2} } \\ \sqrt{36 + 9} \\ \sqrt{45} \\ 6.708$

Length of CA:-

$\sqrt{ {(1 - 4)}^{2} + {( - 4 - 1)}^{2} } \\ \sqrt{ {(3)}^{2} + {( - 5)}^{2} } \\ \sqrt{9 + 25} \\ \sqrt{34} \\ 5.831$

Perimeter of ∆ABC → AB+BC+CA

$8.544 + 6.708 + 5.831 \\ 21.083$

Required Area

• $\longmapsto\;\bold{19.5\;sq.units}$

Required Perimeter

• $\longmapsto\;\bold{21.083\;units}$

Extra Information ℹ️

Semiperimeter: s = 10.542

Angle ∠ A = α = 51.52° = 0.899 rad

Angle ∠ B = β = 42.879° = 0.748 rad

Angle ∠ C = γ = 85.601° = 1.494 rad

Inradius: r = 1.85

Circumradius: R = 4.285

$\boxed{\sf{Area\;of\; Triangle = \frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|}}$

$\boxed{\sf{Distance\; between\;two\; points = \sqrt{(x_2-x_1)²+(y_2-y_1)²}}}$

$\tt{@MrMonarque}$

Hope It Helps You ✌️