Math / Geometry

QuestionFind the value of $x$ .

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes the angles of this triangle work. Watch out! Remember all the angles in a triangle add up to 180 degrees, and be careful with those fractions!

STEP 2

1. Simplify the angle expressions.
2. Set up the equation.
3. Solve for x.

STEP 3

Alright, let's simplify these angle expressions!
We've got $2(x + 5 \frac{1}{2})$ .
Let's convert that mixed number to an improper fraction: $5 \frac{1}{2}$ is the same as $\frac{5 \cdot 2 + 1}{2} = \frac{11}{2}$ .

STEP 4

So, the first angle becomes $2(x + \frac{11}{2})$ .
Distributing the 2, we get $2 \cdot x + 2 \cdot \frac{11}{2} = 2x + 11$ .
Awesome!

STEP 5

Now, let's tackle $6(x - \frac{3}{2})$ .
Distributing the 6, we have $6 \cdot x - 6 \cdot \frac{3}{2} = 6x - \frac{18}{2} = 6x - 9$ .
Fantastic!

STEP 6

We know the angles in a triangle add up to 180 degrees.
Our simplified angles are $2x + 11$ and $6x - 9$ .
The problem doesn't give us the third angle, but since the image shows it's a right triangle, we know the third angle is 90 degrees!

STEP 7

So, our equation is $2x + 11 + 6x - 9 + 90 = 180$ .
This says all our angles added together equal 180!

STEP 8

Let's combine like terms: $2x + 6x$ gives us $8x$ , and $11 - 9 + 90$ gives us $92$ .
So, our equation is now $8x + 92 = 180$ .

STEP 9

Now, we want to isolate $x$ .
Let's subtract 92 from both sides: $8x + 92 - 92 = 180 - 92$ , which simplifies to $8x = 88$ .

STEP 10

Finally, let's divide both sides by 8 to get $x$ all by itself: $\frac{8x}{8} = \frac{88}{8}$ .
This gives us $x = 11$ !

STEP 11

$x = 11$ !
We found it!

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