Trigonometry Problem: Ratio Of Tangents In A Right Triangle

Hey guys, ever stumbled upon a tricky trigonometry problem that just makes you scratch your head? Well, today we're diving deep into one that involves right triangles, angle tangents, and a bit of geometric intuition. Get ready to sharpen those pencils and flex those brain muscles because this one's a real gem!

Understanding the Problem

Before we jump into solving it, let's break down the problem step by step. We're given a right triangle ABC, where the right angle is at B ([AB] ⊥ [BC]). Now, we have some equal segments: |EC| = |DE| = |BD|. This is a crucial piece of information because it hints at some special relationships within the triangle. We're also given the measures of three angles: m(∠ADB) = z, m(∠AEB) = y, and m(∠ACB) = x. Our ultimate goal? To find the value of the ratio (tan x + tan z) / (tan x + tan y). We have two options to choose from: A) 7/5 and B) 8/5. So, how do we tackle this? The key here is to visualize and draw the triangle and mark every information provided. This is how you are going to easily see the relationships between the angles and sides.

The first thing we can observe is that since $|EC| = |DE| = |BD|$, this means that the points D and E divide the side BC into three equal parts. Let's denote the length of each of these parts as a, so we have $|BD| = |DE| = |EC| = a$. This equal division gives us a good starting point for expressing the tangent values of the angles in terms of the sides of the triangle. Remember, the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. For example, in triangle ABC, tan x = |AB| / |BC|. We'll use this basic trigonometric definition to express tan x, tan y, and tan z in terms of the sides of the triangle and then try to find the desired ratio. It might seem a bit daunting now, but trust me, as we break it down, it will all start making sense. Grab your calculators, guys; we're about to embark on a trigonometric adventure!

Setting Up the Ratios

Now, let’s put our trigonometry hats on and start setting up the ratios. This is where the magic happens, guys! Remember that tangent is all about "opposite over adjacent." We're going to express tan x, tan y, and tan z in terms of the sides of our trusty right triangle ABC. Let's start with the easiest one. In triangle ABC, angle x is ∠ACB. So, tan x is simply the length of the opposite side (|AB|) divided by the length of the adjacent side (|BC|). We already know that |BC| is made up of three equal segments, each of length a, so |BC| = 3a. Let's denote the length of |AB| as h (for height). This gives us our first equation: tan x = h / (3a). See? Not too scary, right?

Next up, let's tackle tan z. Angle z is ∠ADB. To find tan z, we need to consider the triangle ABD. In this triangle, the side opposite to angle z is |AB|, which we've already called h. The adjacent side is |BD|, which is just a. Therefore, tan z = h / a. We're on a roll now! Only one more tangent to go. Now for the slightly trickier one, tan y. Angle y is ∠AEB. We need to look at triangle ABE. The opposite side to angle y is still |AB|, our h. But the adjacent side is |BE|, which is made up of two segments, so |BE| = 2a. That means tan y = h / (2a). Alright, we've got all our players on the field! We've successfully expressed tan x, tan y, and tan z in terms of h and a. Now comes the fun part: plugging these values into our target ratio and seeing what simplifications we can make. This is like a trigonometric puzzle, and we're about to fit the pieces together. Let's keep going!

Calculating the Ratio

Okay, guys, the moment we've been waiting for! We've got all the ingredients; now it's time to bake our trigonometric cake. We're trying to find the value of (tan x + tan z) / (tan x + tan y). Remember those expressions we found for tan x, tan z, and tan y? Let's bring them back into the spotlight:

• tan x = h / (3a)
• tan z = h / a
• tan y = h / (2a)

Now, let's substitute these into our ratio. We get: [(h / (3a)) + (h / a)] / [(h / (3a)) + (h / (2a))]. Eek! Looks a bit messy, doesn't it? But don't worry; we're going to clean it up with some good old-fashioned algebra. The first thing we can do is factor out h from both the numerator and the denominator. This gives us: [h(1 / (3a) + 1 / a)] / [h(1 / (3a) + 1 / (2a))]. Notice that h is now a common factor in both the top and bottom, so we can cancel them out! This simplifies our ratio to: (1 / (3a) + 1 / a) / (1 / (3a) + 1 / (2a)). Much better, right?

Now, let's get rid of those fractions within fractions. To do this, we need to find a common denominator for the terms in the numerator and the terms in the denominator. In the numerator, the common denominator is 3a, and in the denominator, it's 6a. Let's rewrite the fractions with these common denominators: Numerator: (1 / (3a)) + (1 / a) = (1 / (3a)) + (3 / (3a)) = 4 / (3a). Denominator: (1 / (3a)) + (1 / (2a)) = (2 / (6a)) + (3 / (6a)) = 5 / (6a). So our ratio now looks like this: (4 / (3a)) / (5 / (6a)). To divide fractions, we multiply by the reciprocal of the second fraction: (4 / (3a)) * ((6a) / 5). Now we can cancel out common factors again. We can cancel a from the numerator and denominator, and we can simplify 6/3 to 2. This leaves us with: (4 * 2) / 5 = 8 / 5. Boom! We've got our answer. The ratio (tan x + tan z) / (tan x + tan y) is equal to 8/5.

Conclusion

Alright, guys, we did it! We successfully navigated through this trigonometric maze and emerged victorious with the answer: 8/5 (Option B). Give yourselves a pat on the back! This problem was a fantastic exercise in applying trigonometric definitions, setting up ratios, and using algebraic manipulation to simplify expressions. Remember, the key to tackling tricky problems like this is to break them down into smaller, more manageable steps.

First, we made sure we understood the problem completely and visualized the triangle with all the given information. Then, we expressed tan x, tan y, and tan z in terms of the sides of the triangle. Next, we substituted these expressions into the target ratio and used algebraic techniques to simplify the fraction. Finally, we arrived at our answer. Trigonometry can sometimes feel like learning a new language, but with practice and perseverance, you can become fluent in it. Keep practicing, keep exploring, and don't be afraid to tackle those challenging problems head-on. You've got this! And hey, if you ever get stuck, remember that breaking it down and taking it step by step is always a winning strategy. Until next time, happy problem-solving, guys!