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

The sum is 6

Explanation:

Given

`Vertices: (0,1), (3,4), (4,3), (3,0)`

`Perimeter: a√(2) + b√(10)`

Required

`a + b`

The first step is to name each points, as follows

`A: (0,1)\\B: (3,4)\\C: (4,3)\\D: (3,0)`

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;

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

Calculating distance AB

`A: (0,1)\\B: (3,4)`

Here,

`x_1 = 0; x_2 = 3`

`y_1 = 1; y_2 = 4`

`AB = √((0 - 3)^2 + (1 - 4)^2)`

`AB = √((- 3)^2 + (-3)^2)`

`AB = √(9+9)`

`AB = √(18)`

Expand 18 as 9 * 2

`AB = √(9 * 2)`

Split surds

`AB = √(9) * √(2)`

Take square root of 9

`AB = 3 * √(2)`

`AB = 3 √(2)`

Calculating distance BC

`B: (3,4)\\C: (4,3)`

Here,

`x_1 = 3; x_2 = 4`

`y_1 = 4; y_2 = 3`

`BC = √((3 - 4)^2 + (4 - 3)^2)`

`BC = √((-1)^2 + (1)^2)`

`BC = √(1 + 1)`

`BC = √(2)`

Calculating distance CD

`C: (4,3)\\D: (3,0)`

Here,

`x_1 = 4; x_2 = 3`

`y_1 = 3; y_2 = 0`

`CD = √((4 - 3)^2 + (3 - 0)^2)`

`CD = √((1)^2 + (3 )^2)`

`CD = √(1 + 9)`

`CD = √(10)`

Calculating distance CD

`D: (3,0)\\A: (0,1)`

Here,

`x_1 = 3; x_2 = 0`

`y_1 = 0; y_2 = 1`

`DA = √((3 - 0)^2 + (0 - 1)^2)`

`DA = √((3)^2 + (- 1)^2)`

`DA = √(9 + 1)`

`DA = √(10)`

At this point, the perimeter can then be calculated

`Perimeter = AB + BC + CD + DA`

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

`Perimeter = 4 √(2) + 2√(10)`

From the given parameters;

`Perimeter: a√(2) + b√(10)`

This implies that;

`a√(2) + b√(10) = 4 √(2) + 2√(10)`

By comparison;

a = 4 and b = 2

Hence;

`a + b = 4 + 2`

`a +b = 6`