For the non-primitive triple (9, 12, 15) that has a common factor greater than 1, dividing by their greatest common factor gives which Baudhāyana triple that still satisfies ?
  • (3, 4, 5)
  • (6, 8, 10)
  • (9, 12, 15)
  • (4, 5, 6)
How can one construct a square having double the area of a given square? Choose the method that always works for any square.
  • Double the length of each side of the given square
  • Construct a square on the diagonal of the given square
  • Rotate the given square by 45 degrees without changing side length
  • Join the midpoints of two adjacent sides of the square
Why does the new dotted square have double the area of the original square? The figure shows a square with a tilted (dotted) square built on its diagonal.
  • Because each side of the dotted square is twice as long as a side of the original square
  • Because the original square can be partitioned into two equal triangles, while the dotted square can be partitioned into four congruent triangles of the same size
  • Because the dotted square’s perimeter is twice that of the original square
  • Because the dotted square and the original square have equal diagonals
Why should the extensions of the vertical and horizontal sides of the original square pass through the vertices of the dotted square?
  • They are angle bisectors of the dotted square and therefore pass through the opposite vertices
  • They are parallel to the sides of the dotted square and therefore intersect them at right angles
  • They are perpendicular bisectors of the original square’s sides and so avoid the dotted square
  • They lie along the medians of the dotted square and therefore meet at its center only
All the small triangles formed by the horizontal and vertical guidelines in the construction are congruent to each other. Which justification best explains this?
  • Side–Angle–Side congruence using equal halves of sides and equal included angles
  • Angle–Side–Angle congruence using two right angles and any side
  • Hypotenuse–Leg congruence for obtuse triangles
  • No triangles are congruent because their orientations differ
After cutting two identical squares of paper as shown and placing pieces 5, 6, 7, and 8 around Square 1, what figure do you obtain?
  • A rectangle with the same area as Square 1
  • A square with double the area of Square 1
  • A square with half the area of Square 1
  • A rhombus with the same area as Square 1
We want a square whose area is half that of a given square. Which construction achieves this?
  • Draw a tilted square by joining the midpoints of the sides of the original square
  • Draw a square on the diagonal of the original square
  • Halve the side length of the original square
  • Rotate the original square by 90 degrees
Why is the smaller inside square half the area of the larger square?
  • Its diagonal equals the larger square’s side length
  • It consists of exactly two of the four congruent triangles formed by drawing the east–west and north–south lines through the large square’s center
  • Its perimeter is half the perimeter of the larger square
  • Its side length is half that of the larger square
Using paper folding, how can you make a square whose area is half the area of the first square?
  • Fold each corner to the center so creases pass through the midpoints of the sides; the diamond PQRS formed is the required square
  • Fold the paper in half along one side and cut along the fold
  • Fold along a diagonal and cut off one triangular half
  • Fold each side outward to extend the square
Will a square having half the side length of the original square have half the area?
  • Yes; halving side length halves the area
  • No; halving side length makes the area one-fourth of the original, so four such squares fill the original
  • Yes; two such smaller squares will fill the original
  • No; halving side length makes the area one-third of the original
Why is PQRS a square, and why is its area half that of the original paper after folding all four corners to the center?
  • PQRS has equal sides and right angles formed by perpendicular creases; it is composed of two of the four congruent corner triangles, giving half the area
  • PQRS is a rectangle with unequal sides; it covers one-third of the paper
  • PQRS is a rhombus without right angles; it covers three-fourths of the paper
  • PQRS is any quadrilateral; its area equals the whole paper
Is less than or greater than ?
  • Less than
  • Equal to
  • Greater than
Will we ever get a number with a terminating decimal representation whose square is ?
  • Yes, a terminating decimal can square to
  • No, a terminating decimal cannot square to
Can be expressed as a fraction , where and are counting numbers?
  • Yes, equals a fraction of counting numbers
  • No, cannot be expressed as a fraction of counting numbers
Two identical square papers are cut as shown into triangular pieces labeled 1–4. Can these pieces be arranged to create a square with double the area of either square?
  • Yes, the pieces can be arranged into such a square
  • No, the pieces cannot form such a square
The length of the two equal sides of an isosceles right triangle is . Choose bounds on the hypotenuse that have at least one digit after the decimal point.
  • Between and
  • Between and
  • Between and
  • Between and
What construction allows two squares of different sizes to be combined to make a larger square whose area equals the sum of the areas of the two smaller squares?
  • Make a right-angled triangle whose perpendicular sides are the sidelengths of the two squares and take the square on its hypotenuse
  • Place the smaller square entirely inside the larger and connect opposite corners
  • Rotate one square by and overlay it on the other
  • Join the squares edge-to-edge to form a rectangle and square the rectangle’s area
Why does Baudhāyana’s method work?
  • Because in a right triangle, the area of the square on the hypotenuse equals the sum of the areas of the squares on the two perpendicular sides
  • Because any two squares can be rearranged to have equal areas by cutting and pasting
  • Because the perimeter of the larger square equals the sum of the perimeters of the smaller squares
  • Because the hypotenuse is always equal to the sum of the other two sides
When the two original squares are the same size, does Baudhāyana’s method agree with the earlier method used to combine two identical squares into a bigger square?
  • Yes, it agrees; the square on the diagonal of a right triangle with equal legs matches the earlier construction
  • No, it does not agree; the resulting square has a different area
  • It agrees only when each square has area
  • It cannot be determined without measurement
In the construction where four congruent right triangles labeled T, U, W, and X are arranged around a central quadrilateral V to form a 4‑sided figure over the hypotenuse, which statement best explains why all four sides of the new 4‑sided figure are equal in length?
  • Because T, U, W, and X are congruent, the segments they touch along the boundary are equal, making all sides equal
  • Because V is a square, each outer side automatically equals its diagonal
  • Because the areas of T, U, W, and X are different, their side lengths must compensate to be equal
  • Because the hypotenuse of each triangle is parallel to the base of the large square
0 h : 0 m : 1 s

Answered Not Answered Not Visited Correct : 0 Incorrect : 0