Final answer:
The angle formed by the shorter side and a diagonal of a rectangle with sides 25 cm and 8 cm is found using right triangle trigonometry and is approximately 72 degrees.
Step-by-step explanation:
To find the measure of the angle formed by the shorter side and a diagonal of the rectangle, we can make use of right triangle trigonometry. The rectangle has sides of 25 cm and 8 cm, so if we draw a diagonal, we create two right-angled triangles within the rectangle. Each triangle has sides of 25 cm, 8 cm, and the diagonal as the hypotenuse.
We are interested in finding the angle between the shorter side (8 cm) and the diagonal. This angle can be found using the cosine function, which is adjacent over hypotenuse in a right-angled triangle. Thus, cos(θ) = 8 cm / Diagonal. However, to use the cosine function, we first need to find the length of the diagonal which can be calculated by the Pythagorean theorem. The square of the hypotenuse (the diagonal) is equal to the sum of the squares of the other two sides. Using this, we get Diagonal² = 25² + 8² = 625 + 64 = 689, and thus Diagonal ≈ √689 cm.
To find the angle, we take the inverse cosine of the ratio 8/√689 and get θ = cos⁻¹(8 / √689).
By calculating this value, we find that θ ≈ cos⁻¹(8 / 26.2488) ≈ cos⁻¹(0.3048) which equals approximately 72.5°. Therefore, the measure of the angle formed by the shorter side and a diagonal of the rectangle is around 72 degrees.