5 - 4 = 12 - h; $\frac{4}{12} = \frac{h}{5}$; 5h = 4·12; $\frac{12}{4} = \frac{h}{5}$; $\frac{5}{4} = \frac{h}{12}$; $\frac{5}{4} = \frac{12}{h}$

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Accepted Answer

To determine which equations are equivalent (assuming this is about similar triangles or proportion problems, likely involving solving for $ h $ in a proportional relationship), we analyze each equation:

### 1. Analyze $ \boldsymbol{\frac{5}{4} = \frac{h}{12}} $ (the selected one)
Cross - multiply: $ 5\times12 = 4\times h$, so $ 60 = 4h$ or $ 5h=4\times12$ (if we rearrange, but let's check other equations).

### 2. Analyze $ \boldsymbol{\frac{5}{4}=\frac{12}{h}} $
Cross - multiply: $ 5h=4\times12$, which is the same as the cross - product of $ \frac{5}{4}=\frac{h}{12} $ (since $ \frac{5}{4}=\frac{h}{12}\Rightarrow5\times12 = 4h$ and $ \frac{5}{4}=\frac{12}{h}\Rightarrow5h = 4\times12$, these are equivalent in terms of the product relationship). Also, $ 5h = 4\times12$ is a direct equation from cross - multiplying $ \frac{5}{4}=\frac{h}{12}$ or $ \frac{5}{4}=\frac{12}{h}$.

### 3. Analyze $ \boldsymbol{\frac{12}{4}=\frac{h}{5}} $
Simplify $ \frac{12}{4}=3$, so $ 3=\frac{h}{5}\Rightarrow h = 15$. Cross - multiply $ \frac{12}{4}=\frac{h}{5}\Rightarrow12\times5=4h\Rightarrow60 = 4h$, which is the same as the cross - product of $ \frac{5}{4}=\frac{h}{12}$ (since $ 5\times12=4h$ is also $ 60 = 4h$).

### 4. Analyze non - proportional equations
- $ 5 - 4=12 - h$: Simplifies to $ 1=12 - h\Rightarrow h = 11$, which is a linear equation of a different form (not proportional) and will give a different value of $ h$ compared to the proportional ones.
- $ \frac{4}{12}=\frac{h}{5}$: Cross - multiply gives $ 4\times5 = 12h\Rightarrow20 = 12h\Rightarrow h=\frac{5}{3}$, which is different from the values obtained from the proportional equations related to $ \frac{5}{4}=\frac{h}{12}$.

If we assume the problem is about similar triangles (a common context for such proportions), the equations that are equivalent (represent the same proportional relationship) are:

- $ \frac{5}{4}=\frac{h}{12}$
- $ \frac{5}{4}=\frac{12}{h}$
- $ 5h = 4\times12$
- $ \frac{12}{4}=\frac{h}{5}$

If we are to find the equation equivalent to $ \frac{5}{4}=\frac{h}{12}$, the equivalent equations are:

- $ \frac{5}{4}=\frac{12}{h}$ (because cross - multiplying both gives $ 5h = 4\times12$)
- $ 5h=4\times12$ (direct cross - product)
- $ \frac{12}{4}=\frac{h}{5}$ (cross - multiplying gives $ 12\times5 = 4h$, which is the same as $ 5\times12=4h$)

If we take the selected equation $ \frac{5}{4}=\frac{h}{12}$, the equivalent equations (by cross - multiplication or rearrangement) are $ \frac{5}{4}=\frac{12}{h}$, $ 5h = 4\times12$, and $ \frac{12}{4}=\frac{h}{5}$.

If we solve $ \frac{5}{4}=\frac{h}{12}$ for $ h$:

## Step 1: Cross - multiply
From $ \frac{5}{4}=\frac{h}{12}$, we use the cross - multiplication property of proportions ($ \frac{a}{b}=\frac{c}{d}\Rightarrow ad = bc$). So, $ 5\times12=4\times h$.

## Step 2: Solve for $ h$
$ 60 = 4h$. Divide both sides by 4: $ h=\frac{60}{4}=15$.

If we solve $ \frac{5}{4}=\frac{12}{h}$:

## Step 1: Cross - multiply
$ 5h=4\times12$.

## Step 2: Solve for $ h$
$ 5h = 48\Rightarrow h=\frac{48}{5}=9.6$? Wait, no, wait. Wait, $ 4\times12 = 48$, so $ 5h=48\Rightarrow h=\frac{48}{5}=9.6$? But this contradicts the previous result. Wait, I made a mistake.

Wait, $ \frac{5}{4}=\frac{h}{12}$: cross - multiply is $ 5\times12=4h\Rightarrow60 = 4h\Rightarrow h = 15$.

$ \frac{5}{4}=\frac{12}{h}$: cross - multiply is $ 5h=4\times12\Rightarrow5h = 48\Rightarrow h=\frac{48}{5}=9.6$. These are not equivalent. I made a mistake in the earlier analysis.

Let's start over. Let's assume the proportion is from similar triangles where the ratio of corresponding sides is equal. Suppose we have two similar triangles, one with sides 5 and 4, and the other with sides $ h$ and 12 (or 12 and $ h$).

The correct proportion for similar triangles (corresponding sides) should have the ratios of corresponding sides equal. So if one triangle has sides $ a = 5$, $ b = 4$ and the other has sides $ c=h$, $ d = 12$, then $ \frac{a}{b}=\frac{c}{d}$ (if $ a$ corresponds to $ c$ and $ b$ corresponds to $ d$) or $ \frac{a}{d}=\frac{b}{c}$ (if $ a$ corresponds to $ d$ and $ b$ corresponds to $ c$).

Case 1: $ \frac{5}{4}=\frac{h}{12}$ ( $ a = 5$, $ b = 4$, $ c = h$, $ d = 12$)
Cross - multiply: $ 5\times12=4h\Rightarrow60 = 4h\Rightarrow h = 15$.

Case 2: $ \frac{12}{4}=\frac{h}{5}$ ( $ a = 12$, $ b = 4$, $ c = h$, $ d = 5$)
Cross - multiply: $ 12\times5=4h\Rightarrow60 = 4h\Rightarrow h = 15$. Ah, here we go. So $ \frac{12}{4}=\frac{h}{5}$ is equivalent to $ \frac{5}{4}=\frac{h}{12}$ because both give $ h = 15$.

And $ 5h=4\times12$: $ 5h = 48\Rightarrow h=\frac{48}{5}=9.6$ is wrong. Wait, no, $ 5\times12 = 60$, so $ 4h=60\Rightarrow h = 15$, and $ 5h=4\times12$ is $ 5h = 48$ which is incorrect. I see my mistake. $ \frac{5}{4}=\frac{h}{12}$ cross - multiplies to $ 5\times12=4h$, not $ 5h = 4\times12$. So $ 5\times12 = 4h$ is $ 60 = 4h$, so $ h = 15$.

$ \frac{12}{4}=\frac{h}{5}$ cross - multiplies to $ 12\times5=4h\Rightarrow60 = 4h\Rightarrow h = 15$, so it is equivalent.

$ \frac{5}{4}=\frac{12}{h}$ cross - multiplies to $ 5h=4\times12\Rightarrow5h = 48\Rightarrow h = 9.6$, which is not equivalent.

And $ 5 - 4=12 - h$ gives $ h = 11$, not equivalent.

$ \frac{4}{12}=\frac{h}{5}$ cross - multiplies to $ 4\times5=12h\Rightarrow20 = 12h\Rightarrow h=\frac{5}{3}$, not equivalent.

So the equations equivalent to $ \frac{5}{4}=\frac{h}{12}$ are $ \frac{12}{4}=\frac{h}{5}$ (because they both give $ h = 15$ when solved) and also $ 5\times12 = 4h$ (or $ 60 = 4h$).

If we consider the equation $ \frac{5}{4}=\frac{h}{12}$, the correct equivalent equations (by the property of proportions $ \frac{a}{b}=\frac{c}{d}\Leftrightarrow\frac{b}{a}=\frac{d}{c}$ or $ \frac{a}{c}=\frac{b}{d}$):

From $ \frac{5}{4}=\frac{h}{12}$, we can also write $ \frac{12}{4}=\frac{h}{5}$ (by swapping the means or the extremes in the proportion). The proportion $ \frac{a}{b}=\frac{c}{d}$ can be rewritten as $ \frac{d}{b}=\frac{c}{a}$ (swapping $ a$ and $ d$)? Wait, the property of proportions: if $ \frac{a}{b}=\frac{c}{d}$, then $ ad = bc$, and also $ \frac{a}{c}=\frac{b}{d}$ (alternendo property). So from $ \frac{5}{4}=\frac{h}{12}$, by alternendo, $ \frac{5}{h}=\frac{4}{12}$, which is not one of the given equations. But $ \frac{12}{4}=\frac{h}{5}$ is obtained by inverting the first ratio and then using alternendo? Wait, $ \frac{5}{4}=\frac{h}{12}$ can be written as $ \frac{4}{5}=\frac{12}{h}$ (invert both ratios), and then by alternendo $ \frac{4}{12}=\frac{5}{h}$, which is not helpful.

But when we solve $ \frac{5}{4}=\frac{h}{12}$ and $ \frac{12}{4}=\frac{h}{5}$, we get $ h = 15$ for both, so they are equivalent. Also, $ 5\times12=4h$ (which is $ 60 = 4h$) is the cross - product form of $ \frac{5}{4}=\frac{h}{12}$.

If we are to find the equations that are equivalent to $ \frac{5}{4}=\frac{h}{12}$ (the one with the blue check), the equivalent ones are:

- $ \frac{12}{4}=\frac{h}{5}$
- $ 5\times12 = 4h$ (or $ 60 = 4h$ or $ 5h=4\times12$ was a mistake, it's $ 5\times12 = 4h$)

But since the problem seems to be about identifying equivalent proportional equations (maybe from a similar - triangles context), the equations equivalent to $ \frac{5}{4}=\frac{h}{12}$ are $ \frac{12}{4}=\frac{h}{5}$ and $ 5h = 4\times12$ was incorrect, the correct cross - product is $ 5\times12=4h$.

However, based on the initial selection of $ \frac{5}{4}=\frac{h}{12}$, the equivalent equation (by cross - multiplication and solving) is $ \frac{12}{4}=\frac{h}{5}$ (since both give $ h = 15$) and the equation $ 5h = 4\times12$ was a miscalculation on my part earlier. The correct cross - product for $ \frac{5}{4}=\frac{h}{12}$ is $ 5\times12=4h$, which is $ 60 = 4h$, and for $ \frac{12}{4}=\frac{h}{5}$ it's $ 12\times5=4h$, which is also $ 60 = 4h$, so they are equivalent.

If we are to solve $ \frac{5}{4}=\frac{h}{12}$:

# Explanation:
## Step 1: Apply cross - multiplication
For the proportion $ \frac{5}{4}=\frac{h}{12}$, using the property of proportions $ \frac{a}{b}=\frac{c}{d}\Rightarrow ad = bc$, we have $ 5\times12 = 4\times h$.
$$5\times12 = 4h$$
## Step 2: Solve for $ h$
Simplify the left - hand side: $ 60 = 4h$. Then divide both sides by 4: $ h=\frac{60}{4}=15$.
# Answer:
$ h = 15$ (and the equivalent equations are $ \frac{12}{4}=\frac{h}{5}$, $ 5\times12 = 4h$)

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QUESTION IMAGE

Question

Explanation:

1. Analyze \( \boldsymbol{\frac{5}{4} = \frac{h}{12}} \) (the selected one)

2. Analyze \( \boldsymbol{\frac{5}{4}=\frac{12}{h}} \)

3. Analyze \( \boldsymbol{\frac{12}{4}=\frac{h}{5}} \)

4. Analyze non - proportional equations

Step 1: Cross - multiply

Step 2: Solve for \( h\)

Step 1: Cross - multiply

Step 2: Solve for \( h\)

Step 1: Apply cross - multiplication

Step 2: Solve for \( h\)

Answer: