Question 1: Explain two defining properties of a kite in terms of its diagonals and angles.

Answer: A kite has two pairs of adjacent equal sides, one diagonal is the perpendicular bisector of the other, and one pair of opposite angles are equal.

Question 2: Calculate the lengths of diagonals if one diagonal measures 8 cm and the other diagonal bisects it at right angles, forming four right triangles with hypotenuses 5 cm each.

Answer: Diagonal lengths: 8 cm and approximately 6.4 cm (using Pythagoras).

Question 3: Construct a kite where two adjacent sides are 6 cm and 10 cm, with the diagonals intersecting at right angles. Label the diagonals.

Answer: Student draws and labels the diagonals crossing at right angles, with the sides of 6 cm and 10 cm meeting accordingly.

Question 4: Prove that the angles between the equal sides of a kite are equal.

Answer: Using the properties of the diagonals and angles in a kite, show that the angles between the pairs of equal sides are equal due to symmetry and congruent triangles.

Question 5: Given a kite with diagonals of 12 cm and 16 cm, where diagonals intersect at right angles, find the area of the kite.

Answer: The area = (1/2) x 12 cm x 16 cm = 96 cm².

Question 6: A kite has one diagonal measuring 10 cm and the other diagonal bisects it at right angles, forming four right triangles with hypotenuses of 13 cm. Find the length of the other diagonal.

Answer: Diagonal length: approximately 24 cm (using Pythagoras).

Question 7: In a real-world scenario, a kite-shaped window has diagonals of 4 m and 6 m. If the window is made of glass with a uniform thickness, estimate the amount of glass used, considering the window as a kite-shaped area.

Answer: Area = (1/2) x 4 m x 6 m = 12 m², so approximately 12 m² of glass.

Question 8: Identify and correct the mistake: A student claims that in a kite, the diagonals are equal in length.

Answer: Correction: In a kite, the diagonals are perpendicular, but they are not necessarily equal in length.

Question 9: Challenge: Given a kite where the lengths of two adjacent sides are 7 cm and 9 cm, and the diagonals intersect at right angles, derive an expression for the length of the diagonal that bisects the other, in terms of the side lengths.

Answer: Using Pythagoras in the right triangles formed, the length of the bisected diagonal can be expressed as √(side² - (half of the other diagonal)²).

Question 10: Evaluate how the properties of a kite can be used to solve for unknown angles in a problem with given side lengths and diagonal measures.

Answer: Apply the symmetry and perpendicular bisector properties to establish relationships between angles and diagonals, then use trigonometry or the Law of Cosines to find unknown angles.