Question

question 19 of 40
write an expression that represents the perimeter of the figure and simplif
the perimeter is
(blank 1,)
blank 1. options
\\(\mathrm{cm}^3\\)
\\(\mathrm{cm}^2\\)
\\(\mathrm{cm}\\)

Explanation:

Step1: Recall Perimeter Formula

Perimeter of a triangle is the sum of its three sides. Let the sides be \( a = \frac{x + 1}{x} \), \( b = x^{\frac{1}{2}} \) (wait, no, the second side is \( x^{\frac{1}{2}} \)? Wait, the original figure: first side \( \frac{x + 1}{x} \) cm, second side \( x^{\frac{1}{2}} \) cm? Wait, no, looking at the image: the second side is \( x^{\frac{1}{2}} \)? Wait, no, the user's image: the three sides are \( \frac{x + 1}{x} \) cm, \( x^{\frac{1}{2}} \) cm (wait, no, the notation: \( \frac{1}{2}x \)? Wait, no, the text: " \( \frac{1}{2}x \) cm"? Wait, maybe a typo. Wait, the three sides: first side \( \frac{x + 1}{x} \) cm, second side \( \frac{1}{2}x \) cm, third side \( \frac{x - 3}{2} \) cm. Wait, let's re-express:

Wait, the three sides are:
1. \( \frac{x + 1}{x} \) cm
2. \( \frac{1}{2}x \) cm (assuming \( x^{\frac{1}{2}} \) is a typo, maybe \( \frac{1}{2}x \))
3. \( \frac{x - 3}{2} \) cm

Wait, no, the original problem: the three sides are \( \frac{x + 1}{x} \), \( x^{\frac{1}{2}} \)? No, that can't be. Wait, maybe the second side is \( \frac{1}{2}x \) (i.e., \( \frac{x}{2} \)) and the third is \( \frac{x - 3}{2} \). Let's proceed with the sum of the three sides.

Perimeter \( P = \frac{x + 1}{x} + \frac{x}{2} + \frac{x - 3}{2} \)

Step2: Combine Like Terms

First, combine the last two terms: \( \frac{x}{2} + \frac{x - 3}{2} = \frac{x + x - 3}{2} = \frac{2x - 3}{2} = x - \frac{3}{2} \)

Now, add the first term: \( P = \frac{x + 1}{x} + x - \frac{3}{2} \)

Wait, this seems complicated. Wait, maybe the second side is \( \frac{1}{2}x \) (i.e., \( \frac{x}{2} \)) and the first side is \( \frac{x + 1}{x} \), third is \( \frac{x - 3}{2} \). Wait, maybe I misread the second side. Let's check again. The user's image: the three sides are:

1. \( \frac{x + 1}{x} \) cm
2. \( x^{\frac{1}{2}} \) cm? No, that's \( \sqrt{x} \), but then the third side is \( \frac{x - 3}{2} \). This is confusing. Wait, maybe the second side is \( \frac{1}{2}x \) (i.e., \( \frac{x}{2} \)) and the third is \( \frac{x - 3}{2} \). Let's sum them:

\( \frac{x + 1}{x} + \frac{x}{2} + \frac{x - 3}{2} \)

Combine the last two terms: \( \frac{x + x - 3}{2} = \frac{2x - 3}{2} = x - \frac{3}{2} \)

Now, add \( \frac{x + 1}{x} \):

\( P = \frac{x + 1}{x} + x - \frac{3}{2} \)

Wait, this doesn't simplify nicely. Maybe the second side is \( \frac{1}{2}x \) (i.e., \( \frac{x}{2} \)) and the first side is \( \frac{x + 1}{x} \), third is \( \frac{x - 3}{2} \). Wait, maybe the original problem has a typo, but let's proceed with the sum of the three sides as given.

Wait, maybe the three sides are:

1. \( \frac{x + 1}{x} \)
2. \( \frac{x}{2} \)
3. \( \frac{x - 3}{2} \)

So sum: \( \frac{x + 1}{x} + \frac{x}{2} + \frac{x - 3}{2} \)

Combine the last two terms: \( \frac{x + x - 3}{2} = \frac{2x - 3}{2} = x - \frac{3}{2} \)

Now, add \( \frac{x + 1}{x} \):

\( P = \frac{x + 1}{x} + x - \frac{3}{2} \)

Wait, this is not matching. Maybe the second side is \( \frac{1}{2}x \) (i.e., \( \frac{x}{2} \)) and the first side is \( \frac{x + 1}{x} \), third is \( \frac{x - 3}{2} \). Wait, maybe the problem is to simplify the sum:

Wait, let's re-express the first term: \( \frac{x + 1}{x} = 1 + \frac{1}{x} \)

Then the sum is \( 1 + \frac{1}{x} + \frac{x}{2} + \frac{x - 3}{2} \)

Combine \( \frac{x}{2} + \frac{x - 3}{2} = \frac{2x - 3}{2} = x - \frac{3}{2} \)

So total: \( 1 + \frac{1}{x} + x - \frac{3}{2} = x - \frac{1}{2} + \frac{1}{x} \)

But this seems odd. Wait, maybe the second side is \( \frac{1}{2}x \) (i.e., \( \frac{x}{2} \)) and the third is \( \frac{x - 3}{2} \), and the first is \( \frac{x + 1}{x} \). Alternatively, maybe the three sides are \( \frac{x + 1}{x} \), \( \frac{x}{2} \), \( \frac{x - 3}{2} \). Let's compute the sum:

\( \frac{x + 1}{x} + \frac{x}{2} + \frac{x - 3}{2} = \frac{x + 1}{x} + \frac{x + x - 3}{2} = \frac{x + 1}{x} + \frac{2x - 3}{2} = \frac{x + 1}{x} + x - \frac{3}{2} \)

This is the perimeter expression. Now, simplifying:

\( \frac{x + 1}{x} + x - \frac{3}{2} = 1 + \frac{1}{x} + x - \frac{3}{2} = x - \frac{1}{2} + \frac{1}{x} \)

But the options for Blank 1 are \( \text{cm}^3 \), \( \text{cm}^2 \), \( \text{cm} \). Perimeter is a length, so units are cm. So the perimeter expression's unit is cm.

Wait, maybe the problem is to find the perimeter expression and then the unit. The perimeter is the sum of the three sides, so the expression is \( \frac{x + 1}{x} + \frac{x}{2} + \frac{x - 3}{2} \) (simplified as above) and the unit is cm.

Answer:

The perimeter expression (after simplifying) is \( x + \frac{1}{x} - \frac{1}{2} \) (or equivalent) and the unit is \(\text{cm}\), so Blank 1 is \(\text{cm}\).