Find Angle Y In Triangle XYZ: A Step-by-Step Solution
Hey everyone! Let's dive into a fascinating geometry problem today. We're going to break down a triangle puzzle, figure out some angle measurements, and most importantly, have fun while doing it. So, buckle up and let's get started!
The Triangle XYZ Challenge
Our challenge involves a triangle, which we'll call XYZ. Now, triangles, as you might recall from geometry class, are those three-sided shapes where the angles always add up to 180 degrees. This is a crucial piece of information, so keep it in your back pocket.
The problem gives us some clues about the relationships between the angles in our triangle. Specifically, we know:
- Angle Y is 2/7 of the measure of angle Z.
- Angle Y is 20% of the measure of angle X.
Our mission, should we choose to accept it (and of course, we do!), is to find out the measure of angle Y in degrees. It sounds a bit like a puzzle, right? That's because it is! We just need to translate these word clues into math and solve for our unknown.
Breaking Down the Clues
Let's take these clues one at a time and turn them into mathematical expressions. This is like translating from English to Math, which can sound intimidating, but it's totally manageable. Think of it as building a bridge between words and numbers.
Our first clue says: "Angle Y is 2/7 of the measure of angle Z." In math terms, we can write this as:
Y = (2/7) * Z
See? Not so scary. We've just said that the measure of angle Y is equal to two-sevenths times the measure of angle Z. We're using letters to represent the angles, which is a common practice in algebra. It's like giving a nickname to something so we can talk about it more easily.
Now, let's tackle the second clue: "Angle Y is 20% of the measure of angle X." Okay, so 20% might sound a little tricky, but remember that percent means "out of one hundred." So, 20% is the same as 20/100, which simplifies to 1/5. We can also write it as a decimal, 0.20. All these forms are just different ways of saying the same thing.
So, we can translate our clue into the following equation:
Y = (1/5) * X
Or, equivalently:
Y = 0.20 * X
We've now turned both our clues into neat little mathematical equations. This is a huge step, guys! We've transformed the word problem into something we can actually work with using algebra.
Setting Up the Equations
Now that we've decoded our clues, we have two equations:
Y = (2/7) * ZY = (1/5) * X
Remember that key piece of information we tucked away at the beginning? The fact that the angles in a triangle add up to 180 degrees? That gives us our third equation:
X + Y + Z = 180
We now have a system of three equations with three unknowns (X, Y, and Z). This is like having a treasure map with three different routes to the treasure. We just need to follow the right path to find our answer.
Systems of equations might seem daunting, but they're actually quite powerful tools. They allow us to relate different variables and solve for unknowns by combining information. There are several ways to solve systems of equations, but for this problem, we'll use a method called substitution.
The Substitution Solution
The substitution method is like playing a game of connect-the-dots. We use one equation to express one variable in terms of others, and then substitute that expression into another equation. This reduces the number of unknowns and makes the problem simpler.
Looking at our equations, we notice that both equation 1 and equation 2 express Y in terms of other variables. This is perfect for substitution! Let's start by using equation 1 (Y = (2/7) * Z) and equation 2 (Y = (1/5) * X). Since both of these expressions are equal to Y, they must also be equal to each other. This gives us:
(2/7) * Z = (1/5) * X
We've now created a new equation that relates X and Z directly. This is like finding a secret passage that connects two rooms in our problem. Let's solve this equation for one of the variables. To avoid fractions, let's multiply both sides of the equation by the least common multiple of 7 and 5, which is 35:
35 * (2/7) * Z = 35 * (1/5) * X
This simplifies to:
10Z = 7X
Now, let's solve for X by dividing both sides by 7:
X = (10/7) * Z
Great! We've expressed X in terms of Z. This is like finding a key that unlocks a door. Now, we can substitute this expression for X into our third equation (X + Y + Z = 180). But first, let's also substitute the expression for Y from equation 1 (Y = (2/7) * Z) into our third equation. This will give us an equation with only one unknown, Z. This is the home stretch, guys!
So, substituting X and Y into equation 3, we get:
(10/7) * Z + (2/7) * Z + Z = 180
Solving for Z
We now have an equation with only Z as the unknown. Let's combine the terms with Z. To do this, we need a common denominator, which is 7. We can rewrite Z as (7/7) * Z. So, our equation becomes:
(10/7) * Z + (2/7) * Z + (7/7) * Z = 180
Adding the fractions, we get:
(19/7) * Z = 180
To solve for Z, we multiply both sides of the equation by the reciprocal of 19/7, which is 7/19:
Z = 180 * (7/19)
Z = 1260 / 19
Z ≈ 66.32 degrees
We've found the measure of angle Z! This is a major victory! We're one step closer to finding angle Y.
Finding Angle Y
Now that we know the measure of angle Z, we can easily find the measure of angle Y using equation 1: Y = (2/7) * Z. Substituting the value we found for Z, we get:
Y = (2/7) * (1260 / 19)
Y = 2520 / (7 * 19)
Y = 2520 / 133
Y = 18.95 degrees
Therefore, The measure of angle Y is 18.95 degrees, rounded to two decimal places.
Double-Checking Our Work
Before we declare victory, it's always a good idea to double-check our work. Let's plug our values for Y and Z back into equation 2 (Y = (1/5) * X) and equation 3 (X + Y + Z = 180) to see if they hold true.
First, let's find X using equation 2:
18.95 = (1/5) * X
Multiplying both sides by 5, we get:
X = 94.75 degrees
Now, let's plug our values for X, Y, and Z into equation 3:
94.75 + 18.95 + 66.32 = 180.02
The sum is very close to 180 degrees! The slight difference is due to rounding errors. This confirms that our solution is correct.
Conclusion
We did it, guys! We successfully navigated the clues, set up equations, used substitution, and solved for the measure of angle Y. This problem shows how we can use algebra to solve real-world (or at least triangle-world) puzzles. Remember, math is like a toolbox filled with powerful tools. The more tools you learn to use, the more challenges you can overcome. So keep practicing, keep exploring, and most importantly, keep having fun with math!
