Question

question 1 - 6
fill in the blanks using the available answer choices
the perimeter of the triangle is approximately units and its area is approximately square units
a (-4,4)
b (10,2)
c (2,-4)
d (3,3)

Explanation:

Step1: Calculate the length of side AB using the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$

For points $A(-4,4)$ and $B(10,2)$:
$AB=\sqrt{(10 + 4)^2+(2 - 4)^2}=\sqrt{14^2+( - 2)^2}=\sqrt{196 + 4}=\sqrt{200}=10\sqrt{2}\approx14.14$

Step2: Calculate the length of side BC

For points $B(10,2)$ and $C(2,-4)$:
$BC=\sqrt{(2 - 10)^2+( - 4 - 2)^2}=\sqrt{( - 8)^2+( - 6)^2}=\sqrt{64+36}=\sqrt{100}=10$

Step3: Calculate the length of side AC

For points $A(-4,4)$ and $C(2,-4)$:
$AC=\sqrt{(2 + 4)^2+( - 4 - 4)^2}=\sqrt{6^2+( - 8)^2}=\sqrt{36 + 64}=\sqrt{100}=10$

Step4: Calculate the perimeter P

$P=AB + BC+AC\approx14.14+10 + 10=34.14$

Step5: Calculate the area A using Heron's formula or base - height method. Here we use base - height. Let base be AB and height be the length of CD.

Length of CD: For points $C(2,-4)$ and $D(3,3)$
$CD=\sqrt{(3 - 2)^2+(3 + 4)^2}=\sqrt{1+49}=\sqrt{50}=5\sqrt{2}\approx7.07$
$A=\frac{1}{2}\times AB\times CD=\frac{1}{2}\times10\sqrt{2}\times5\sqrt{2}=\frac{1}{2}\times100 = 50$

Answer:

The perimeter is approximately 34.14 units and the area is approximately 50 square units.