Answer:
The sum is 6
Step-by-step explanation:
Given
[tex]Vertices: (0,1), (3,4), (4,3), (3,0)[/tex]
[tex]Perimeter: a\sqrt{2} + b\sqrt{10}[/tex]
Required
[tex]a + b[/tex]
The first step is to name each points, as follows
[tex]A: (0,1)\\B: (3,4)\\C: (4,3)\\D: (3,0)[/tex]
Next is to calculate the distance between each consecutive point
We'll calculate the distance AB, BC, CD and DA
Distance between points is calculated as thus;
[tex]d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}[/tex]
Calculating distance AB
[tex]A: (0,1)\\B: (3,4)[/tex]
Here,
[tex]x_1 = 0; x_2 = 3[/tex]
[tex]y_1 = 1; y_2 = 4[/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]
Expand 18 as 9 * 2
[tex]AB = \sqrt{9 * 2}[/tex]
Split surds
[tex]AB = \sqrt{9} * \sqrt{2}[/tex]
Take square root of 9
[tex]AB = 3 * \sqrt{2}[/tex]
[tex]AB = 3 \sqrt{2}[/tex]
Calculating distance BC
[tex]B: (3,4)\\C: (4,3)[/tex]
Here,
[tex]x_1 = 3; x_2 = 4[/tex]
[tex]y_1 = 4; y_2 = 3[/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 distance CD
[tex]C: (4,3)\\D: (3,0)[/tex]
Here,
[tex]x_1 = 4; x_2 = 3[/tex]
[tex]y_1 = 3; y_2 = 0[/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]
Calculating distance CD
[tex]D: (3,0)\\A: (0,1)[/tex]
Here,
[tex]x_1 = 3; x_2 = 0[/tex]
[tex]y_1 = 0; y_2 = 1[/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]
At this point, the perimeter can then be calculated
[tex]Perimeter = AB + BC + CD + DA[/tex]
[tex]Perimeter = 3 \sqrt{2} + \sqrt{2}\ + \sqrt{10} + \sqrt{10}[/tex]
[tex]Perimeter = 4 \sqrt{2} + 2\sqrt{10}[/tex]
From the given parameters;
[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;
a = 4 and b = 2
Hence;
[tex]a + b = 4 + 2[/tex]
[tex]a +b = 6[/tex]
