Right Triangle Side And Angle Calculation

Hey guys! Let's dive into a fascinating mathematical puzzle involving right triangles. We've got a classic scenario: a right triangle with a hypotenuse measuring 10 cm and one leg (a cateto, in Portuguese) measuring 6 cm. Our mission? To find the length of the other leg (which we'll call 'y') and to determine the measure of the angle 'x' opposite the 6 cm leg. Buckle up, because we're about to embark on a journey through the Pythagorean Theorem and trigonometric functions! Let's jump in and solve this together.

Unearthing the Missing Side: The Pythagorean Theorem to the Rescue

The Pythagorean Theorem, a cornerstone of geometry, provides the key to unlocking the length of the missing leg. Remember this gem? It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs). In mathematical terms, this translates to: a² + b² = c², where 'c' represents the hypotenuse and 'a' and 'b' represent the legs. This is a fundamental concept in understanding the relationships within right triangles. Now, let's apply this powerful tool to our problem.

We know the hypotenuse (c) is 10 cm and one leg (let's say 'a') is 6 cm. Our goal is to find the other leg, 'y' (which we'll call 'b' in the formula). Plugging the values into the Pythagorean Theorem, we get: 6² + y² = 10². Let's simplify this: 36 + y² = 100. To isolate y², we subtract 36 from both sides: y² = 100 - 36, which gives us y² = 64. Now, to find 'y', we take the square root of both sides: y = √64. And there you have it! The square root of 64 is 8, so the length of the missing leg, y, is 8 cm. Isn't it amazing how a simple theorem can reveal such crucial information? But our adventure doesn't end here; we still have to conquer the mystery of angle 'x'. Understanding and applying the Pythagorean Theorem is not just about solving problems; it's about grasping the inherent beauty and order within geometric shapes. This principle extends beyond the classroom, influencing fields like architecture, engineering, and even art. So, when you see a triangle, remember the power hidden within its sides.

Deciphering the Angle: Trigonometry to the Rescue

Now that we've found the missing side, it's time to tackle the second part of our quest: determining the measure of angle 'x'. For this, we'll need to call upon the forces of trigonometry. Trigonometry is the branch of mathematics that deals with the relationships between the angles and sides of triangles. Specifically, we'll be using trigonometric functions – sine, cosine, and tangent – to help us. These functions relate an angle of a right triangle to the ratios of its sides. Let's focus on the sine function first, as it directly connects the angle 'x' to the sides we know.

The sine of an angle (sin) is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In our case, the side opposite angle 'x' is 6 cm, and the hypotenuse is 10 cm. Therefore, sin(x) = opposite/hypotenuse = 6/10 = 0.6. Now, to find the measure of angle 'x' itself, we need to use the inverse sine function, also known as arcsine (denoted as sin⁻¹ or arcsin). This function essentially asks,