Answer: The correct options are;
*The quadrilaterals cannot be placed such that each occupies one quarter of the circle because the vertices of parallelogram 1 do not form right angles
**The quadrilaterals cannot be placed such that each occupies one quarter of the circle because the vertices of parallelogram 4 do not form right angles.
Step-by-step explanation: What James is trying to do is quite simple which is, he wants to place four quadrilaterals inside a circle and he wants the vertices to touch one another at the center of the circle without having to overlap.
This is possible and quite simple, provided all the quadrilaterals have right angles (90 degrees). This is because the center of the circle measures 360 degrees and we can only have four vertices placed there without overlapping only if they all measure 90 degrees each (that is, 90 times 4 equals 360).
We can now show whether or not all four parallelograms have right angles by applying the Pythagoras' theorem to each of them. Note that James has cut the shapes in such a way that the hypotenuse (diagonal) and the other two legs have already been given in the question. As a reminder, the Pythagoras' theorem is given as,
AC² = AB² + BC² Where AC is the hypotenuse (diagonal) and AB and BC are the other two legs. The experiment would now be as follows;
Quadrilateral 1;
20² = 12² + 15²
400 = 144 + 225
400 ≠ 369
Therefore the vertices of parallelogram 1 do not form a right angle
Quadrilateral 2;
34² = 16² + 30²
1156 = 256 + 900
1156 = 1156
Therefore the vertices of parallelogram 2 forms a right angle
Quadrilateral 3;
29² = 20² + 21²
841 = 400 + 441
841 = 841
Therefore the vertices of parallelogram 3 forms a right angle
Quadrilateral 4;
26² = 18² + 20²
676 = 324 + 400
676 ≠ 724
Therefore the vertices of parallelogram 4 do not form a right angle
The results above shows that only two of the parallelograms cut out have right angles (like a proper square or rectangle for instance), while the other two do not have right angles.
Therefore, the correct option are as follows;
The quadrilaterals cannot be placed such that each occupies one quarter of the circle because the vertices of parallelogram 1 do not form right angles.
The quadrilaterals cannot be placed such that each occupies one quarter of the circle because the vertices of parallelogram 4 do not form right angles.
