Question

the slope between a and c is the same as the slope between b and d. choose the correct justification for this fact. △ace and △bdf are congruent, so the ratio \\( \frac { e c } { a e } \\) is equal to the ratio \\( \frac { b f } { f d } \\). the slope between a and c and the slope between b and d are both equal to \\( \frac { 2 } { 3 } \\). △ace and △bdf are congruent, so the ratio \\( \frac { e c } { a e } \\) is equal to the ratio \\( \frac { f d } { b f } \\). the slope between a and c and the slope between b and d are both equal to \\( \frac { 2 } { 3 } \\). △ace and △bdf are similar, so the ratio \\( \frac { a e } { e c } \\) is equal to the ratio \\( \frac { f d } { b f } \\). the slope between a and c and the slope between b and d are both equal to \\( \frac { 3 } { 2 } \\). △ace and △bdf are similar, so the ratio \\( \frac { a e } { e c } \\) is equal to the ratio \\( \frac { b f } { f d } \\). the slope between a and c and the slope between b and d are both equal to \\( \frac { 3 } { 2 } \\).

Explanation:

1. First, identify the triangles: $\triangle ACE$ and $\triangle BDF$ are right triangles (since they are formed by vertical and horizontal segments on the grid) and share the same angle with the line $AC$ (or $BD$), so they are similar (by AA similarity, as both have a right angle and the same non - right angle).
2. For the slope of a line, the slope $m=\frac{\text{rise}}{\text{run}}$. In $\triangle ACE$, the rise is the vertical change (length of $AE$) and the run is the horizontal change (length of $EC$). In $\triangle BDF$, the rise is the vertical change (length of $BF$) and the run is the horizontal change (length of $FD$).
3. For similar triangles, the ratios of corresponding sides are equal. So $\frac{AE}{EC}=\frac{BF}{FD}$.
4. Now, calculate the slope. Let's find the coordinates of points. From the grid, if we assume the coordinates: Let's say point $A$ is at $(-6, - 7)$ and point $C$ (we can find the change). The vertical change (rise) between $A$ and a point (say the intersection with the grid lines) and horizontal change (run). But more simply, for the line, the slope is $\frac{\text{vertical change}}{\text{horizontal change}}$. If we calculate the slope between $A$ and $C$, from the grid, the vertical change (rise) is $3$ units (from $y=-7$ to $y = - 4$) and horizontal change (run) is $2$ units (from $x=-6$ to $x=-4$)? Wait, no, actually, looking at the line, the slope should be $\frac{3}{2}$. Wait, let's re - examine. If we take two points on the line, say $A(-6,-7)$ and $B(-2,-2)$ (assuming $B$'s coordinates). The vertical change (rise) from $A$ to $B$ is $(-2)-(-7)=5$? No, maybe my initial coordinate assumption is wrong. But the key is that for similar triangles, the ratio of rise to run (slope) should be equal. The correct ratio for the slope (rise over run) for the line in the grid, when we calculate the vertical and horizontal changes, we find that the slope is $\frac{3}{2}$. And since the triangles are similar, the ratio of the vertical side (rise) to horizontal side (run) in each triangle is equal, so $\frac{AE}{EC}=\frac{BF}{FD}$ and the slope is $\frac{3}{2}$. The first two options say the triangles are congruent, but congruent triangles would have equal side lengths, but here the triangles are similar (same shape, different size) because the line has a constant slope, so the triangles are similar, not congruent. The third option has the wrong ratio for the sides (it has $\frac{AE}{EC}=\frac{FD}{BF}$ which is incorrect, it should be $\frac{AE}{EC}=\frac{BF}{FD}$). So the fourth option is correct as it states the triangles are similar, the ratio of corresponding sides (rise over run) are equal, and the slope is $\frac{3}{2}$.

Answer:

The fourth option: $\boldsymbol{\triangle ACE}$ and $\boldsymbol{\triangle BDF}$ are similar, so the ratio $\boldsymbol{\frac{AE}{EC}}$ is equal to the ratio $\boldsymbol{\frac{BF}{FD}}$. The slope between $A$ and $C$ and the slope between $B$ and $D$ are both equal to $\boldsymbol{\frac{3}{2}}$.