In a 45-degree right-angled triangle, the perpendicular is 8, the base is 8, and the hypotenuse is 8√2, determined through trigonometric functions and the Pythagorean theorem.
In a right-angled triangle, trigonometric functions can be used to relate the angles to the sides. Given a right-angled triangle with an angle θ = 45 degrees, we can establish the relationships among the sides. Let the perpendicular side be represented by 8, the base by x, and the hypotenuse by y.
The trigonometric function relevant to this scenario is the tangent function, defined as tan(θ) = opposite / adjacent. In this case, tan(45 degrees) = 8 / x.
Since tan(45 degrees) is equal to 1, the equation simplifies to 1 = 8 / x, and solving for x yields x = 8.
Now that we know x, we can use the Pythagorean theorem to find y, which states that y^2 = perpendicular^2 + base^2. Substituting in the values, we get y^2 = 8^2 + 8^2, which simplifies to y^2 = 128. Taking the square root of both sides, we find y = √128, and further simplification yields y = 8√2.
In summary, for a right-angled triangle with a 45-degree angle, the perpendicular side is 8, the base is 8, and the hypotenuse is 8√2.