Question

1. a scale drawing of the side view of a small wooden bridge is shown below. the architect creates a new drawing with a scale of one grid unit equal to foot. what is the length of the base of the bridge in the new drawing? (image of bridge scale drawing with 2ft indicator) angle 1 is a scale drawing of triangle 2, as shown below. triangle 1 (side 10, side 12) triangle 2 (base 35, side 42, side x) the information shown in these triangles...

Explanation:

First Problem (Bridge Scale Drawing)

Step1: Count grid units for base

Assume each small grid's side (in original) relates to 2ft. First, count the number of grid units the base spans. From the diagram, let's say the base spans 8 grid units (visually, from the left to right mark). Wait, no—wait, the original scale: the small segment (2ft) is 1 grid unit? Wait, the diagram has a "2ft" label for a small segment. Let's re-express:

Wait, the problem says "the architect creates a new drawing with a scale of one grid unit equal to [let's assume the original scale: first, find the original length of the base. Wait, the original drawing: each grid unit (in original) is 2ft? Wait, the "2ft" is a label for a small part. Let's count the base's grid units. Looking at the bridge's base: from the left end to right end, how many grid units? Let's see, the base is from, say, column 1 to column 9 (8 units? Wait, no, the arrow: let's count the number of grid squares the base covers. Let's say the base is 8 grid units long in the original drawing, and each grid unit (original) is 2ft? Wait, no—wait, the new scale is "one grid unit equal to foot" (maybe 1 grid unit = 1ft? Wait, the problem statement might have a typo, but let's proceed. Wait, maybe the original scale: each grid unit (in the given drawing) is 2ft. So first, find the length of the base in the original drawing (in grid units), then convert to the new scale (1 grid unit = 1ft? Or maybe the original scale is 2ft per grid unit, and new scale is 1 grid unit = 1ft? Wait, no—let's re-express:

Wait, the original drawing: the "2ft" is a segment that's 1 grid unit. So original scale: 1 grid unit = 2ft. Now, the base of the bridge in the original drawing: how many grid units? Let's count the base's length. From the left mark to right mark, the base spans 8 grid units (visually, looking at the grid). So original length: 8 grid units 2ft/grid unit = 16ft? No, wait, maybe the base is 8 grid units, and original scale is 2ft per grid unit, so original length is 82=16ft? But the new scale is "one grid unit equal to foot" (maybe 1 grid unit = 1ft). Wait, no—maybe the new scale is 1 grid unit = 1ft, and we need to find the length in the new drawing. Wait, no—first, find the actual length of the base, then convert to new scale.

Wait, the original drawing: the small segment (labeled 2ft) is 1 grid unit. So original scale: 1 grid unit = 2ft. Now, the base of the bridge: count the number of grid units. Let's say the base is 8 grid units (from the left arrow to right arrow). So original length: 8 2ft = 16ft? No, wait, maybe the base is 8 grid units, and each grid unit (original) is 2ft, so length is 82=16ft. Now, the new drawing has a scale of 1 grid unit = 1ft. So in the new drawing, the length would be 16ft / 1ft per grid unit = 16 grid units? Wait, no—maybe I messed up. Wait, maybe the original drawing's grid: each square is 2ft per side. So the base spans, say, 8 squares (width). So original length: 8 * 2ft = 16ft. New scale: 1 grid unit = 1ft, so new length is 16 grid units? But that seems off. Wait, maybe the original scale is 2ft per grid unit, and the base is 8 grid units, so original length is 16ft. New scale: 1 grid unit = 1ft, so new drawing's base length is 16 grid units (since 1 grid unit = 1ft, so 16ft is 16 grid units). But maybe the base is 8 grid units in original, each grid unit is 2ft, so length is 16ft. New scale: 1 grid unit = 1ft, so length in new drawing is 16 grid units (i.e., 16ft, but as grid units, 16). Wait, maybe the correct count: let's look at the bridge's base. The base is from the left end to right end, a…

Step1: Set up proportion for similar triangles

Since Triangle 1 ~ Triangle 2 (similar), corresponding sides are proportional. So:

$\frac{\text{Side of Triangle 1}}{\text{Corresponding Side of Triangle 2}} = \frac{\text{Another Side of Triangle 1}}{\text{Corresponding Side of Triangle 2}}$

Triangle 1: sides 10 (base), 12 (side)

Triangle 2: sides 35 (base), 42 (side), and $x$ (corresponding to 12)

So proportion: $\frac{10}{35} = \frac{12}{x}$? Wait, no—wait, corresponding sides: Triangle 1's base (10) corresponds to Triangle 2's base (35), Triangle 1's side (12) corresponds to Triangle 2's side ($x$)? Wait, no, wait: Triangle 1's side 12 corresponds to Triangle 2's side 42? Wait, no, let's check the labels. Triangle 1: base 10, side 12. Triangle 2: base 35, side 42, and $x$ (the other side). Wait, no, maybe Triangle 1's base (10) corresponds to Triangle 2's base (35), and Triangle 1's side (12) corresponds to Triangle 2's side ($x$), and Triangle 1's other side (let's say) corresponds to 42. Wait, no—similar triangles have corresponding sides in proportion. So:

$\frac{10}{35} = \frac{12}{x} = \frac{\text{other side}}{42}$

Wait, no, let's match the sides. Triangle 1: base 10, side 12. Triangle 2: base 35, side 42, and $x$ (the side corresponding to 12). Wait, no—maybe the sides are:

Triangle 1: 10 (base), 12 (leg)

Triangle 2: 35 (base), 42 (leg), and $x$ (the other leg, corresponding to 12? No, wait, 10 corresponds to 35, 12 corresponds to $x$, and the third side corresponds to 42? Wait, no, 10/35 = 12/x = third side/42. Wait, 10/35 simplifies to 2/7. So 12/x = 2/7 → $x = 12 * 7 / 2 = 42$? Wait, no, that can't be. Wait, maybe I mixed up the correspondence. Let's re-express:

Wait, Triangle 1: sides 10 (base), 12 (side)

Triangle 2: sides 35 (base), 42 (side), $x$ (side corresponding to 12)

Wait, no—if Triangle 1's base (10) corresponds to Triangle 2's base (35), and Triangle 1's side (12) corresponds to Triangle 2's side (42), then the ratio is 10/35 = 12/42? Wait, 10/35 = 2/7, 12/42 = 2/7. Oh! So that's the ratio. So then, the other side of Triangle 1 (let's say, the one corresponding to $x$) would be... Wait, no, the problem is to find $x$. Wait, maybe the sides are:

Triangle 1: 10 (base), 12 (side)

Triangle 2: 35 (base), $x$ (side), 42 (side)

So the proportion is $\frac{10}{35} = \frac{12}{x} = \frac{\text{third side}}{42}$

But since 10/35 = 2/7, then 12/x = 2/7 → $x = 12 * 7 / 2 = 42$? Wait, no, that's not right. Wait, no—wait, 10 corresponds to 35 (ratio 2/7), 12 corresponds to 42 (12*3.5=42, 3.5=35/10), so then the other side of Triangle 1 (let's say, the one we don't see) corresponds to $x$, and 42 corresponds to that. Wait, no, the problem is to find $x$. Wait, maybe the correspondence is:

Triangle 1: side 12, base 10

Triangle 2: side 42, base 35, and side $x$ (corresponding to 12? No, 12 and 42 are corresponding, 10 and 35 are corresponding, so the ratio is 10/35 = 12/42 = 2/7. Then, if there's another side in Triangle 1 (let's say, length $y$) corresponding to $x$ in Triangle 2, but the problem is to find $x$. Wait, maybe I misread the labels. Let's look again:

Triangle 1: base 10, side 12

Triangle 2: base 35, side 42, and $x$ (the side adjacent to the base, corresponding to 12)

Wait, no—let's set up the proportion correctly. Let's let the sides be:

For Triangle 1: $a = 10$, $b = 12$

For Triangle 2: $A = 35$, $B = x$, $C = 42$

Since similar, $\frac{a}{A} = \frac{b}{B} = \frac{c}{C}$

We know $a = 10$, $A = 35$, $b = 12$, $C = 42$

So $\frac{10}{35} = \frac{12}{x}$ → Solve for $x$:

$10x = 35 * 12$

$10x…

Answer:

16 (assuming the base spans 8 grid units in original, 2ft per grid, new scale 1ft per grid)

Second Problem (Similar Triangles)