Solving for Side AB in a Right Triangle

In the realm of geometry, particularly dealing with right triangles, it is often necessary to find the lengths of the sides given certain measurements or angles. The problem presented asks us to determine the length of side AB in a right triangle where the hypotenuse AC is given and the length of side BC is known. The triangle involves trigonometric functions, specifically the tangent, which can be effectively utilized to find the unknown side.

Understanding the Right Triangle

Consider a right triangle ABC with the right angle at B. In such a triangle, one of the sides (AC) is the hypotenuse - the side opposite the right angle and the longest side in the triangle. Here, BC is given as 10 cm, but the length of side AB is unknown. Our goal is to find the length of AB, given that AC is the hypotenuse.

Using Trigonometric Functions

Right triangles and trigonometric functions go hand in hand, particularly the tangent function. The tangent (tan) of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. For angle A, the formula is:

[ tan(A) frac{text{opposite}}{text{adjacent}} frac{BC}{AB} ]

Given the length of BC, if we can determine the angle A, we can solve for AB using the inverse tangent function (also known as arctangent or tan-1). Let’s assume angle A is given or can be determined, which allows us to use the inverse tangent function to solve for AB.

Step-by-Step Solution

Let's go through a step-by-step process to solve for AB.

Determine the Angle A: If the angle A is not directly given, we can use the information about the sides to find it. For instance, if we had the length of the hypotenuse AC, we could use the sine or cosine function to determine angle A, but in this problem, we only have the side BC and need to use the tangent function. Assuming we know angle A, we proceed as follows. Use the Tangent Function: Once angle A is known or a relationship can be derived, we use the tangent function as described above. For instance, if angle A is 60 degrees, then: [ tan(60^circ) sqrt{3} frac{10}{AB} ] Isolate AB: Rearrange the equation to solve for AB: [ AB frac{10}{sqrt{3}} text{ or approximately } 5.77 text{ cm} ] Verify Using the Inverse Tangent Function: If the angle A is not directly given, we can use the inverse tangent function to solve for the angle first: [ A tan^{-1}left(frac{BC}{AB}right) ] Then we can confirm the accuracy of our AB value. For example, if angle A is 60 degrees, we can verify: [ tan^{-1}left(frac{10}{AB}right) 60^circ ] Solving for AB again should yield the same result.

Conclusion

Given the information in the problem, if we know the angle at A, we can use trigonometric functions to find the length of the unknown side. If angle A is given, then using the tangent function and the inverse tangent function, we can determine AB. In the specific case where BC 10 cm and angle A is known to be 60 degrees, the length of AB is approximately 5.77 cm.

This process is fundamental in solving a myriad of problems involving right triangles, demonstrating the power and importance of trigonometry in geometry and various fields of mathematics.