Question

statements reasons 1. r ∥ s given 2. ( m_r = \frac{d - b}{c - 0} = \frac{d - b}{c} ) ( m_s = \frac{0 - a}{c - 0} = -\frac{a}{c} ) application of the slope formula 3. distance from (0, b) to (0, a) equals the distance from (c, d) to (c, 0) definition of parallel lines 4.? application of the distance formula 5. ( m_r = \frac{(b - a) - b}{c} ) substitution property of equality 6. ( m_r = -\frac{a}{c} ) inverse property of addition 7. ( m_r = m_s ) substitution property of equality the table and corresponding image show the proof of the relationship between the slopes of two parallel lines. what is the missing statement in step 4? a. ( c - 0 = a - b ) b. ( d - 0 = b - a ) c. ( c - d = b - a ) d. ( b - c = d - a )

Explanation:

Step1: Recall Distance Formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For vertical segments (same \(x\)-coordinate), the distance is \(|y_2 - y_1|\).

Step2: Analyze Distances

- Distance from \((0,b)\) to \((0,a)\): Since \(x\)-coordinates are 0, distance is \(|b - a|=|a - b|\) (absolute value, but for equality, we can consider \(b - a\) or \(a - b\) with sign depending on direction, but here we use the formula result).
- Distance from \((c,d)\) to \((c,0)\): \(x\)-coordinates are \(c\), so distance is \(|d - 0|=|d|\)? Wait, no, wait. Wait, the vertical distance: for \((0,b)\) to \((0,a)\), the change in \(y\) is \(b - a\) (if \(b>a\)) or \(a - b\) (if \(a>b\)), but the distance is \(|b - a|\). For \((c,d)\) to \((c,0)\), the change in \(y\) is \(d - 0=d\)? No, wait, no—wait, the problem says "distance from \((0,b)\) to \((0,a)\) equals the distance from \((c,d)\) to \((c,0)\)". So applying distance formula:

For \((0,b)\) to \((0,a)\): \(d_1=\sqrt{(0 - 0)^2+(a - b)^2}=|a - b|\) (or \(|b - a|\)).

For \((c,d)\) to \((c,0)\): \(d_2=\sqrt{(c - c)^2+(0 - d)^2}=|d - 0|=|d|\)? Wait, no, that can't be. Wait, maybe I misread the points. Wait, step 3 says "distance from \((0,b)\) to \((0,a)\) equals the distance from \((c,d)\) to \((c,0)\)". So let's compute both distances:

Distance 1 (vertical line \(x = 0\)): between \((0,b)\) and \((0,a)\): \(y\)-difference is \(a - b\) (so distance is \(|a - b|\)).

Distance 2 (vertical line \(x = c\)): between \((c,d)\) and \((c,0)\): \(y\)-difference is \(0 - d=-d\) (or \(d - 0=d\)), distance is \(|d - 0|=|d|\)? No, that's not matching. Wait, maybe the points are \((0,b)\) to \((0,a)\): the vertical distance is \(b - a\) (if we consider direction, but the distance formula gives \(\sqrt{(0 - 0)^2+(a - b)^2}=|a - b|\)). For \((c,d)\) to \((c,0)\): \(\sqrt{(c - c)^2+(0 - d)^2}=|d - 0|=|d|\). But that doesn't seem to connect. Wait, no—wait, maybe the problem has a typo? Wait, no, looking at step 5: \(m_r=\frac{(b - a)-b}{c}\)? Wait, no, step 5 is \(m_r=\frac{(b - a)-b}{c}\)? Wait, no, step 5 is \(m_r=\frac{(b - a)-b}{c}\)? Wait, no, let's look at the slope of \(r\): \(m_r=\frac{d - b}{c}\) (from step 2). Then step 5 substitutes something. Wait, the distance from \((0,b)\) to \((0,a)\) is \(|b - a|\), and from \((c,d)\) to \((c,0)\) is \(|d - 0|=|d|\)? No, that's not. Wait, maybe the vertical distance is \(b - a\) (if \(b\) is above \(a\)) and \(d - 0=d\), but that would mean \(b - a=d\)? No, that's not. Wait, no—wait, the distance formula application in step 4: so the distance from \((0,b)\) to \((0,a)\) is \(|b - a|\), and from \((c,d)\) to \((c,0)\) is \(|d - 0|=|d|\)? No, that's not. Wait, maybe the points are \((0,b)\) to \((0,a)\): the change in \(y\) is \(a - b\) (so distance is \(|a - b|\)), and from \((c,d)\) to \((c,0)\): change in \(y\) is \(0 - d=-d\), so distance is \(|d|\). But that doesn't fit. Wait, maybe I made a mistake. Wait, let's look at the answer options.

Option B: \(d - 0=b - a\). Let's check: distance from \((0,b)\) to \((0,a)\) is \(|b - a|\) (using distance formula: \(\sqrt{(0 - 0)^2+(a - b)^2}=|a - b|\)). Distance from \((c,d)\) to \((c,0)\) is \(\sqrt{(c - c)^2+(0 - d)^2}=|d - 0|=|d|\). But if we apply the distance formula (without absolute value for the expression, considering the actual difference), then:

For \((0,b)\) to \((0,a)\): distance is \(|b - a|=b - a\) (assuming \(b>a\)) or \(a - b\) (if \(a>b\)). For \((c,d)\) to \((c,0)\): distance is \(|d - 0|=d\) (if \(d>0\)) or \(-d\) (if \(d<0\)). But step 3 says they are equal, so \(|b - a|=|d - 0|\). But the application of distance formula (step 4) should be the equation from the formula. So the distance formula for \((0,b)\) to \((0,a)\) is \(\sqrt{(0 - 0)^2+(a - b)^2}=|a - b|\), and for \((c,d)\) to \((c,0)\) is \(\sqrt{(c - c)^2+(0 - d)^2}=|d - 0|\). But the "application of the distance formula" in step 4 should be the equation that comes from calculating those distances. So if we drop the absolute value (since we can consider the signed difference or the equality of the distances as expressions), then:

Distance from \((0,b)\) to \((0,a)\): \(|b - a|=b - a\) (if \(b\geq a\)) or \(a - b\) (if \(a\geq b\)).

Distance from \((c,d)\) to \((c,0)\): \(|d - 0|=d\) (if \(d\geq0\)) or \(-d\) (if \(d<0\)).

But the answer option B is \(d - 0=b - a\), which is \(d=b - a\). Let's see: if we apply the distance formula (for vertical lines, distance is \(|y_2 - y_1|\)), so for \((0,b)\) to \((0,a)\), distance is \(|b - a|\), and for \((c,d)\) to \((c,0)\), distance is \(|d - 0|=|d|\). But the problem states these distances are equal (step 3), so \(|b - a|=|d|\). But step 4 is "application of the distance formula", so we write the distance expressions. For \((0,b)\) to \((0,a)\): distance is \(\sqrt{(0 - 0)^2+(a - b)^2}=|a - b|\), and for \((c,d)\) to \((c,0)\): \(\sqrt{(c - c)^2+(0 - d)^2}=|d|\). But the answer options are linear equations, not square roots. Wait, maybe the vertical segments have the same length, so the difference in \(y\)-coordinates is equal (since \(x\)-coordinates are same, the distance is the absolute value of the difference in \(y\)-coordinates, so the actual difference (with sign) can be equal if we consider direction, but for the formula, when we apply distance formula to vertical lines, the distance is \(|y_2 - y_1|\), so the equation from the distance formula (after squaring, but since it's vertical, we can skip the square root for the equality of distances: \((y_2 - y_1)^2=(y_4 - y_3)^2\) if \(x_2=x_1\) and \(x_4=x_3\). But since \(x\) is same, the distance is \(|y_2 - y_1|=|y_4 - y_3|\), so \(y_2 - y_1=\pm(y_4 - y_3)\), but in the proof, we need the positive equality (since we are dealing with lengths). So for \((0,b)\) to \((0,a)\): \(y\)-difference is \(a - b\) (or \(b - a\)), and for \((c,d)\) to \((c,0)\): \(y\)-difference is \(0 - d=-d\) (or \(d - 0=d\)). But the answer option B is \(d - 0=b - a\), which is \(d=b - a\). Let's check the slope in step 5: \(m_r=\frac{(b - a)-b}{c}\). Wait, \(m_r\) was \(\frac{d - b}{c}\) (from step 2). If \(d=b - a\), then substituting \(d\) into \(m_r\): \(\frac{(b - a)-b}{c}=\frac{-a}{c}\), which matches step 6 (\(m_r=-\frac{a}{c}\)) and step 7 (\(m_r=m_s\) since \(m_s=-\frac{a}{c}\) from step 2). So let's verify:

If step 4 is \(d - 0=b - a\) (i.e., \(d=b - a\)), then:

- Distance from \((0,b)\) to \((0,a)\): \(|b - a|\) (since \(x\) same, distance is \(|b - a|\)).
- Distance from \((c,d)\) to \((c,0)\): \(|d - 0|=|d|=|b - a|\), which matches step 3.

Now, check the options:

Option A: \(c - 0=a - b\) → \(c=a - b\). Not related to vertical distance.

Option B: \(d - 0=b - a\) → \(d=b - a\). This gives the relationship between \(d\) and \(b - a\), which is needed for substituting into the slope formula (step 5: \(m_r=\frac{(b - a)-b}{c}=\frac{-a}{c}\), which is the slope of \(s\) from step 2 (\(m_s=-\frac{a}{c}\))).

Option C: \(c - d=b - a\) → \(c=d + b - a\). Not vertical distance.

Option D: \(b - c=d - a\) → \(b - d=c - a\). Not vertical distance.

So the missing statement in step 4 (application of distance formula) is \(d - 0=b - a\) (i.e., \(d=b - a\)), which is option B.

Answer:

B. \( d - 0 = b - a \)