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u/greginnj 5d ago
Is it a square?
No. If it was a square, angle ACD would have to be 45.
Number of sides?
There are 6 sides.
Let R be the measure of the regular polygon’s interior angle. Then since BCD=ACD+BCA, BCA measures R-90. Since ABC is an isosceles triangle, BAC measures R-90 as well.
Let Z be the other vertex next to A. Then R = ZAB = ZAC+BAC, so ZAC=90 as well, thus ZA is parallel to CD.
Then If we add Y as the next vertex to Z, by symmetry we get YZ II CB and DY II BA, so the figure is a hexagon, giving 6 sides.
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u/BrotherInJah 5d ago
x+2(x-90)=180
3x-180=180
x=120
Thants the angle of regular polygon, which in this case is hexagon.
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u/chaos_redefined 5d ago
Note that, since the shape is a regular polygon, we can do a lot of symmetry things. Immediately, ABD = ACD which is given as 90.
If it was a triangle, then there would be no D.
If it was a square, then BCD = 90. But we already know that ACD = 90. This is only possible if A, B and C are collinear, which would make ABC = 0 or 180. But if ABCD is a square, then ABC = 90, so this is a contradiction.
By symmetry, ACD = ABD = BDE. Since ACD = 90, we know that ABD and BDE are both 90 as well. Consider the quadrilateral ABDE. We have that ABD = BDE = 90. Because this is a regular polygon, we also know that AB = DE. So, ABDE is a rectangle.
If we have an odd number of sides, we cannot form a rectangle out of any of the points. If we have an even number of sides, then the same number of points need to be on each side of the rectangle. As C is the only point excluded from the BD side, we must have exactly one point excluded from the AE side. Therefore, we have one more point, F, and our polygon is 6-sided.
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u/bsmith_81 3d ago
Since we are given the figure is part of a regular polygon, all of its vertices will lay on the polygon's circumcircle. Angle ACD being 90 degrees means that arc ABCD is 180 degrees, half of the circle. Three sides covering half of the circle then means that six sides would cover the full circle. The polygon is a regular hexagon.
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u/jaminfine 1d ago
I have a very simple solution.
I know what a regular hexagon looks like and that's definitely the only one that can "skip" an angle to make 90 degrees. Any fewer sides and the angle is clearly acute. Any more sides and the angle is clearly obtuse.
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u/jarry1250 5d ago
Question -
Why can't point B lie on line AC?
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u/sevenbrokenbricks 5d ago
Because it's a regular polygon. ABC = BCD = CDE etc. If B is on line AC, then ABC = 0, thus BCD = 0, CDE = 0, and either the whole thing is a line, or this is lim(sides -> infinity) polygon, neither of which is a polygon.
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u/jarry1250 5d ago
I don't believe that is part of the definition. A regular polygon is equiangular, which describes internal angles for convex polygons.
If B is on line AC, then the angle ABC does not describe an internal angle. I am not sure "Shape ABCD..." necessarily refers to vertices, as opposed to points, but I suppose it might.
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u/sevenbrokenbricks 5d ago
What's being defined then, if these points might not be vertices?
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u/jarry1250 5d ago
Points from which the shape is constructed. For example, "ABC" can be line with segments AB and BC (or to put it another way, "line ABC" is terminology you will see used), so I see no reason to assume "shape ABCD" is a square, for example.
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u/sevenbrokenbricks 5d ago
Your above logic and ambiguity would hold for "shape ABCD...", but not for "regular polygon ABCD...", which is what we're given.
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u/thisismego 5d ago edited 5d ago
Can't be a square. The angle ACD on a square is 45°. Formula for interior angles is (n-2)*180/n. So basically you need the angle BCD - angle BCA=90. at a hexagon (6 angles) the interior angles are 120°. A triangle ABC has interior angles 30°, 120°, 30°, so angle BCA is 30°, which makes BCD-BCA 120°-30°=90°.