A quadrilateral has vertices at $(0,1)$, $(3,4)$, $(4,3)$ and $(3,0)$. Its perimeter can be expressed in the form $a\sqrt2+b\sqrt{10}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$?

a + b = 12

Step-by-step explanation:

Given

Quadrilateral;

Vertices of (0,1), (3,4) (4,3) and (3,0)

Required

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;

Substitute these values in the formula above

Calculating BC

Here, we consider B = (3,4); C = (4,3)

Here,

Substitute these values in the formula above

Calculating CD

Here, we consider C = (4,3); D = (3,0)

Here,

Substitute these values in the formula above

Lastly;

Calculating DA

Here, we consider C = (4,3); D = (3,0)

Here,

Substitute these values in the formula above

The addition of the values of distances AB, BC, CD and DA gives the perimeter of the quadrilateral

Recall that

This implies that

By comparison

Divide both sides by

By comparison

Divide both sides by

Hence,

a + b = 2 + 10

a + b = 12

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