a rectangle is 80 cm long and 20 cm wide. find, to the nearest degree, the acute angle formed at the intersections of the diagonals.

Well, if you're looking for an acute angle, you won't find one at the intersections of the diagonals of a rectangle. The intersection point of the diagonals of a rectangle forms right angles, not acute ones. So, to the nearest degree, the answer would be 90 degrees. But don't worry, rectangles can be a little square sometimes!

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

Step 1: Calculate the length of the diagonals. This can be done using the Pythagorean theorem.
- Diagonal 1 (d1) can be found using the formula: d1 = √(length^2 + width^2)
Substituting the given values:
d1 = √(80^2 + 20^2) cm
= √(6400 + 400) cm
= √(6800) cm
≈ 82.46 cm (rounded to two decimal places)

- Diagonal 2 (d2) is the same length as diagonal 1.
Therefore, d2 ≈ 82.46 cm

Step 2: Calculate the tangent of the acute angle.
- The tangent can be found using the formula: tan(θ) = length / width
- In this case, tan(θ) = length of shorter side / length of longer side
- Substituting the given values, we get: tan(θ) ≈ 20 / 80
≈ 0.25

Step 3: Find the acute angle using the inverse tangent function (arctan or tan^-1).
- Using a calculator, find arctan(0.25) ≈ 14.04 degrees (rounded to two decimal places).

Therefore, the acute angle formed at the intersections of the diagonals in the rectangle is approximately 14.04 degrees.

To find the acute angle formed at the intersections of the diagonals of a rectangle, we can use a trigonometric ratio, specifically the inverse tangent (arctan) function. Here's how you can calculate it step by step:

Step 1: Calculate the length of the diagonal.
- In a rectangle, the diagonals are equal in length.
- Use the Pythagorean theorem to find the length of the diagonal (D).
D^2 = Length^2 + Width^2
D^2 = 80^2 + 20^2
D^2 = 6400 + 400
D^2 = 6800
D ≈ √6800
D ≈ 82.462 cm (rounded to three decimal places)

Step 2: Find the acute angle.
- Divide the width by the length of the diagonal to determine the tangent of the angle.
Tan(angle) = Width / Length
Tan(angle) = 20 / 80
Tan(angle) = 0.25

- Now, use the inverse tangent function (arctan) to find the angle.
angle = arctan(0.25)
angle ≈ 14.04 degrees (rounded to two decimal places)

Therefore, to the nearest degree, the acute angle formed at the intersections of the diagonals is approximately 14 degrees.

The diagonal forms a hypotenuse with a length sqrt[80^2 + 20^2] = 82.462

The acute angle formed at the corner is arctan 20/80 = 14.036 degrees. Where the diagonals cross, the acute angle is twice that, or 28.072 degrees. Round that to 28 degrees.