James cut out four parallelograms, the dimensions of which are shown below.
Parallelogram 1
length: 12 in.
width: 15 in.
diagonal: 20 in.
Parallelogram 2
length: 16 in.
width: 30 in.
diagonal: 34 in.
Parallelogram 3
length: 20 in.
width: 21 in.
diagonal: 29 in.
Parallelogram 4
length: 18 in.
width: 20 in.
diagonal: 26 in.
James put the parallelograms together so one vertex from each paper exists on a point, as shown in the circle.
4 parallelograms are put together so that one vertex from each paper exists on a point.
Which statement explains whether or not the parallelgrams can be put together so each occupies one-quarter of the area of the circle without overlapping any other pieces? Check all that apply.
The quadrilaterals can be placed such that each occupies one-quarter of the circle.
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 2 do not form right angles.
The quadrilaterals cannot be placed such that each occupies one-quarter of the circle because the vertices of parallelogram 3 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.
The statement "The quadrilaterals can be placed such that each occupies one-quarter of the circle" is false.
The statements "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 2 do not form right angles," "The quadrilaterals cannot be placed such that each occupies one-quarter of the circle because the vertices of parallelogram 3 do not form right angles," and "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" are false.
Therefore, the answer is "None of the above."
try again
The statement "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" is false because parallelograms do not necessarily have right angles.
The statement "The quadrilaterals cannot be placed such that each occupies one-quarter of the circle because the vertices of parallelogram 2 do not form right angles" is true because the diagonals of parallelogram 2 do not form right angles.
The statement "The quadrilaterals cannot be placed such that each occupies one-quarter of the circle because the vertices of parallelogram 3 do not form right angles" is false because parallelograms do not necessarily have right angles.
The statement "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" is false because parallelograms do not necessarily have right angles.
Therefore, the only correct statement is "The quadrilaterals cannot be placed such that each occupies one-quarter of the circle because the vertices of parallelogram 2 do not form right angles."
The areas of the squares formed by the side lengths of the triangles are given for each triangle below. Which triangles are right triangles? Select the three correct answers. (Figures are not drawn to scale.)
3 squares form a triangle. The squares have areas 6 inches squared, 10 inches squared, 8 inches squared.
4 squares form a triangle. The squares have areas 9 inches squared, 25 inches squared, 16 inches squared.
5 squares form a triangle. The squares have areas 25 inches squared, 169 inches squared, 144 inches squared.
6 squares form a triangle. The squares have areas 7 inches squared, 25 inches squared, 24 inches squared.
7 squares form a triangle. The squares have areas 64 inches squared, 225 inches squared, 289 inches squared.
8 squares form a triangle. The squares have areas 10 inches squared, 26 inches squared, 24 inches squared.
Triangles with side lengths of squares that form right triangles satisfy the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Using this information, we can determine which triangles are right triangles:
For the first triangle: 6^2 + 8^2 = 10^2, which satisfies the Pythagorean theorem. Therefore, the first triangle is a right triangle.
For the second triangle: 9^2 + 16^2 ≠ 25^2, so it does not satisfy the Pythagorean theorem. Therefore, the second triangle is not a right triangle.
For the third triangle: 25^2 + 144^2 = 169^2, which satisfies the Pythagorean theorem. Therefore, the third triangle is a right triangle.
For the fourth triangle: 7^2 + 24^2 ≠ 25^2, so it does not satisfy the Pythagorean theorem. Therefore, the fourth triangle is not a right triangle.
For the fifth triangle: 64^2 + 289^2 ≠ 225^2, so it does not satisfy the Pythagorean theorem. Therefore, the fifth triangle is not a right triangle.
For the sixth triangle: 10^2 + 24^2 = 26^2, which satisfies the Pythagorean theorem. Therefore, the sixth triangle is a right triangle.
Based on the above analysis, the three triangles that are right triangles are the first, third, and sixth triangles.
In order to determine whether or not the parallelograms can be put together so each occupies one-quarter of the area of the circle, we need to consider the angles formed by their vertices.
The statement "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" is incorrect. Parallelogram 1 does not need to have right angles in order for the quadrilaterals to be put together in the desired way.
Similarly, the statements "The quadrilaterals cannot be placed such that each occupies one-quarter of the circle because the vertices of parallelogram 2 do not form right angles," "The quadrilaterals cannot be placed such that each occupies one-quarter of the circle because the vertices of parallelogram 3 do not form right angles," and "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" are also incorrect. The angles formed by the vertices of parallelogram 2, parallelogram 3, and parallelogram 4 do not necessarily need to be right angles for the quadrilaterals to be put together in the desired way.
Therefore, the correct statement is "The quadrilaterals can be placed such that each occupies one-quarter of the circle."
To determine whether or not the parallelograms can be put together so each occupies one-quarter of the area of the circle without overlapping any other pieces, we need to check if the vertices of each parallelogram form right angles.
First, let's check each parallelogram:
Parallelogram 1:
The vertices of parallelogram 1 do not necessarily form right angles. Therefore, the statement "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" applies.
Parallelogram 2:
The vertices of parallelogram 2 do not necessarily form right angles. Therefore, the statement "The quadrilaterals cannot be placed such that each occupies one-quarter of the circle because the vertices of parallelogram 2 do not form right angles" applies.
Parallelogram 3:
The vertices of parallelogram 3 do not necessarily form right angles. Therefore, the statement "The quadrilaterals cannot be placed such that each occupies one-quarter of the circle because the vertices of parallelogram 3 do not form right angles" applies.
Parallelogram 4:
The vertices of parallelogram 4 do not necessarily form right angles. Therefore, the statement "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" applies.
Hence, the correct statements 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 2 do not form right angles.
- The quadrilaterals cannot be placed such that each occupies one-quarter of the circle because the vertices of parallelogram 3 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.