Question

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 $b$?

Answer 1

a+b=6

Explanation:

The tutor verified answer is mostly correct however, if you look under both by comperision sections you will see that it is:

`4√(2)` and
`2√(10)` thus the answer is 4+2=6

Answer 2

a + b = 12

Explanation:

Given

Quadrilateral;

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

`Perimeter = a√(2) + b√(10)`

Required

`a + b`

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;

`Distance = √((x_1 - x_2)^2 + (y_1 - y_2)^2)`

`(x_1,y_1) = A(0,1)`

`(x_2,y_2) = B(3,4)`

Substitute these values in the formula above

`Distance = √((x_1 - x_2)^2 + (y_1 - y_2)^2)`

`AB = √((0 - 3)^2 + (1 - 4)^2)`

`AB = √(( - 3)^2 + (-3)^2)`

`AB = √(9+ 9)`

`AB = √(18)`

`AB = √(9*2)`

`AB = √(9)*√(2)`

`AB = 3√(2)`

Calculating BC

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

Here,

`(x_1,y_1) = B (3,4)`

`(x_2,y_2) = C(4,3)`

Substitute these values in the formula above

`Distance = √((x_1 - x_2)^2 + (y_1 - y_2)^2)`

`BC = √((3 - 4)^2 + (4 - 3)^2)`

`BC = √((-1)^2 + (1)^2)`

`BC = √(1 + 1)`

`BC = √(2)`

Calculating CD

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

Here,

`(x_1,y_1) = C(4,3)`

`(x_2,y_2) = D (3,0)`

Substitute these values in the formula above

`Distance = √((x_1 - x_2)^2 + (y_1 - y_2)^2)`

`CD = √((4 - 3)^2 + (3 - 0)^2)`

`CD = √((1)^2 + (3)^2)`

`CD = √(1 + 9)`

`CD = √(10)`

Lastly;

Calculating DA

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

Here,

`(x_1,y_1) = D (3,0)`

`(x_2,y_2) = A (0,1)`

Substitute these values in the formula above

`Distance = √((x_1 - x_2)^2 + (y_1 - y_2)^2)`

`DA = √((3 - 0)^2 + (0 - 1)^2)`

`DA = √((3)^2 + (- 1)^2)`

`DA = √(9 + 1)`

`DA = √(10)`

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

`Perimeter = 3√(2) + √(2) + √(10) + √(10)`

`Perimeter = 4√(2) + 2√(10)`

Recall that

`Perimeter = a√(2) + b√(10)`

This implies that

`a√(2) + b√(10) = 4√(2) + 2√(10)`

By comparison

`a√(2) = 4√(2)`

Divide both sides by
`√(2)`

`a = 4`

By comparison

`b√(10) = 2√(10)`

Divide both sides by
`√(10)`

`b = 2`

Hence,

a + b = 2 + 10

a + b = 12