A rectangle is 80 cm long and 20 cm wide. Find, to the nearest degree, the acute angle formed at the intersection of the diagonals.

take one of the right-angled triangles, call the base angle Ø

tanØ = 20/80 = 1/4
Ø = appr. 14.04°

can you take it from there?
I see pairs of isosceles triangles !

Well, if you're looking for acute angles in a rectangle, you're definitely barking up the wrong tree! Rectangles are all about right angles, not acute angles! It's like asking a clown to do a somersault instead of juggling! But since I'm a helpful clown, I'll tell you that the diagonals of a rectangle always intersect at a right angle, which means the acute angle formed is actually 90 degrees. I hope I didn't leave you feeling obtuse!

To find the acute angle formed at the intersection of the diagonals of a rectangle, we need to use the property that opposite sides of a rectangle are congruent and the diagonals bisect each other.

In a rectangle with length (L) and width (W), the diagonals are equal in length and divide the rectangle into four congruent right triangles.

Let's denote the length of the rectangle as L = 80 cm and the width as W = 20 cm.

Using the Pythagorean theorem, we can find the length of the diagonal (D) using the equation:

D^2 = L^2 + W^2

Substituting the given values:

D^2 = (80 cm)^2 + (20 cm)^2
D^2 = 6400 cm^2 + 400 cm^2
D^2 = 6800 cm^2

Taking the square root of both sides, we find:

D ≈ 82.46 cm

Since the diagonals of a rectangle bisect each other, we have two congruent right triangles formed at the intersection of the diagonals.

To find the acute angle (θ), we can use the trigonometric ratio:

tan(θ) = (L/2) / (W/2)
tan(θ) = (80 cm/2) / (20 cm/2)
tan(θ) = 40 cm / 10 cm
tan(θ) = 4

Using the inverse tangent function (tan⁻¹), we can find the value of θ:

θ ≈ tan⁻¹(4)
θ ≈ 75.96 degrees

Therefore, to the nearest degree, the acute angle formed at the intersection of the diagonals in the rectangle is approximately 76 degrees.

To find the acute angle formed at the intersection of the diagonals of a rectangle, we can use trigonometry.

Step 1: Find the length of the diagonals.
In a rectangle, the length of the diagonals can be found using the Pythagorean theorem.
The formula for the length of a diagonal in a rectangle is:
diagonal = √(length^2 + width^2)

Given that the length is 80 cm and the width is 20 cm, we can substitute these values into the formula:

diagonal = √(80^2 + 20^2)
diagonal = √(6400 + 400)
diagonal = √(6800)
diagonal ≈ 82.462 cm (rounded to three decimal places)

Step 2: Calculate the acute angle formed at the intersection of the diagonals.
Once we have the lengths of the diagonals, we can use trigonometry to find the acute angle. In this case, we'll use the tangent function:

tan(angle) = (length / width)

angle = arctan(length / width)

angle = arctan(80 / 20)
angle ≈ arctan(4)

Using a calculator, we can find that the arctan(4) is approximately 75.96 degrees.

Therefore, to the nearest degree, the acute angle formed at the intersection of the diagonals of the rectangle is 76 degrees.