Answer:
a + b = 12
Step-by-step explanation:
Given
Quadrilateral;
Vertices of (0,1), (3,4) (4,3) and (3,0)
[tex]Perimeter = a\sqrt{2} + b\sqrt{10}[/tex]
Required
[tex]a + b[/tex]
Let the vertices be represented with A,B,C,D such as
A = (0,1); B = (3,4); C = (4,3) and D = (3,0)
To calculate the actual perimeter, we need to first calculate the distance between the points;
Such that:
AB represents distance between point A and B
BC represents distance between point B and C
CD represents distance between point C and D
DA represents distance between point D and A
Calculating AB
Here, we consider A = (0,1); B = (3,4);
Distance is calculated as;
[tex]Distance = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}[/tex]
[tex](x_1,y_1) = A(0,1)[/tex]
[tex](x_2,y_2) = B(3,4)[/tex]
Substitute these values in the formula above
[tex]Distance = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}[/tex]
[tex]AB = \sqrt{(0 - 3)^2 + (1 - 4)^2}[/tex]
[tex]AB = \sqrt{( - 3)^2 + (-3)^2}[/tex]
[tex]AB = \sqrt{9+ 9}[/tex]
[tex]AB = \sqrt{18}[/tex]
[tex]AB = \sqrt{9*2}[/tex]
[tex]AB = \sqrt{9}*\sqrt{2}[/tex]
[tex]AB = 3\sqrt{2}[/tex]
Calculating BC
Here, we consider B = (3,4); C = (4,3)
Here,
[tex](x_1,y_1) = B (3,4)[/tex]
[tex](x_2,y_2) = C(4,3)[/tex]
Substitute these values in the formula above
[tex]Distance = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}[/tex]
[tex]BC = \sqrt{(3 - 4)^2 + (4 - 3)^2}[/tex]
[tex]BC = \sqrt{(-1)^2 + (1)^2}[/tex]
[tex]BC = \sqrt{1 + 1}[/tex]
[tex]BC = \sqrt{2}[/tex]
Calculating CD
Here, we consider C = (4,3); D = (3,0)
Here,
[tex](x_1,y_1) = C(4,3)[/tex]
[tex](x_2,y_2) = D (3,0)[/tex]
Substitute these values in the formula above
[tex]Distance = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}[/tex]
[tex]CD = \sqrt{(4 - 3)^2 + (3 - 0)^2}[/tex]
[tex]CD = \sqrt{(1)^2 + (3)^2}[/tex]
[tex]CD = \sqrt{1 + 9}[/tex]
[tex]CD = \sqrt{10}[/tex]
Lastly;
Calculating DA
Here, we consider C = (4,3); D = (3,0)
Here,
[tex](x_1,y_1) = D (3,0)[/tex]
[tex](x_2,y_2) = A (0,1)[/tex]
Substitute these values in the formula above
[tex]Distance = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}[/tex]
[tex]DA = \sqrt{(3 - 0)^2 + (0 - 1)^2}[/tex]
[tex]DA = \sqrt{(3)^2 + (- 1)^2}[/tex]
[tex]DA = \sqrt{9 + 1}[/tex]
[tex]DA = \sqrt{10}[/tex]
The addition of the values of distances AB, BC, CD and DA gives the perimeter of the quadrilateral
[tex]Perimeter = 3\sqrt{2} + \sqrt{2} + \sqrt{10} + \sqrt{10}[/tex]
[tex]Perimeter = 4\sqrt{2} + 2\sqrt{10}[/tex]
Recall that
[tex]Perimeter = a\sqrt{2} + b\sqrt{10}[/tex]
This implies that
[tex]a\sqrt{2} + b\sqrt{10} = 4\sqrt{2} + 2\sqrt{10}[/tex]
By comparison
[tex]a\sqrt{2} = 4\sqrt{2}[/tex]
Divide both sides by [tex]\sqrt{2}[/tex]
[tex]a = 4[/tex]
By comparison
[tex]b\sqrt{10} = 2\sqrt{10}[/tex]
Divide both sides by [tex]\sqrt{10}[/tex]
[tex]b = 2[/tex]
Hence,
a + b = 2 + 10
a + b = 12
