the measure of the sides of a rectangle are ( 4 √ ̅2 - √ ̅3cm) by ( √ ̅6 + 2 cm). Find the measure of its diagonal.
sqrt(N) = (N)^1/2.
D^2 = (4*2^1/2-3^1/2)^2 + (6^1/2+2)^2 = 35.2
D = 5.933 cm.
To find the measure of the diagonal of a rectangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the other two sides.
Let's call the length of one side of the rectangle "a" and the length of the other side "b", and the diagonal "d". According to the given information, we have:
a = 4√2 - √3 cm
b = √6 + 2 cm
Using the Pythagorean theorem, we can write the equation:
d^2 = a^2 + b^2
Substituting the given values, we have:
d^2 = (4√2 - √3)^2 + (√6 + 2)^2
Simplifying the equation:
d^2 = (16*2 - 8√6√2 + 3) + (6 + 4√6 + 4)
d^2 = 32 - 8√12 + 3 + 10 + 4√6 + 4
d^2 = 49 + 4√6 - 8√12
Now, since the diagonal cannot have a negative length, we take the positive square root of both sides:
d = √(49 + 4√6 - 8√12)
This is the measure of the diagonal of the rectangle.
To find the measure of the diagonal of a rectangle, you can use the Pythagorean theorem. The Pythagorean theorem states that the square of the hypotenuse (in this case, the diagonal) is equal to the sum of the squares of the other two sides.
Let's label the sides of the rectangle:
- Length of the rectangle = 4√2 - √3 cm
- Width of the rectangle = √6 + 2 cm
Now, we need to find the measure of the diagonal.
Using the Pythagorean theorem, we have:
Diagonal^2 = Length^2 + Width^2
Let's substitute the given values:
Diagonal^2 = (4√2 - √3)^2 + (√6 + 2)^2
To simplify the calculation, let's expand each squared term:
Diagonal^2 = (4√2 - √3)(4√2 - √3) + (√6 + 2)(√6 + 2)
Using the FOIL method (first, outer, inner, last), we get:
Diagonal^2 = 16(2) + 4√2(-√3) + 4√2(-√3) + (-√3)(-√3) + 6 + 4√6 + 4√6 + 4
Simplifying further:
Diagonal^2 = 32 - 4√6 - 4√6 + 3 + 6 + 4√6 + 4√6 + 4
Combining like terms:
Diagonal^2 = 49 + 8√6
Finally, we take the square root of both sides to find the measure of the diagonal:
Diagonal = √(49 + 8√6) cm
Therefore, the measure of the diagonal of the rectangle is √(49 + 8√6) cm.