PQRS is a square with side 10 cm. Its diagonals PR and QS intersect at point T. Which of the following correctly describes triangle PTQ formed by the two half-diagonals and the side PQ?
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Worked Solution
Step 1: Use the properties of a square's diagonals.
• The diagonals of a square bisect each other: PT = TR and QT = TS.
• The diagonals of a square are perpendicular: angle PTQ = 90°.
• The diagonals of a square are equal in length.
Step 2: Find the diagonal length.
Diagonal = side × √2 = 10√2 cm
Half-diagonal PT = QT = 5√2 cm
Step 3: Describe triangle PTQ.
• Angle at T = 90° (diagonals are perpendicular) → right-angled
• PT = QT = 5√2 cm → two equal sides → isosceles
• PQ = 10 cm ≠ 5√2 cm → not all sides equal → not equilateral
• Angles at P and Q: since PT = QT and angle T = 90°, angles TPQ = TQP = (180° − 90°) ÷ 2 = 45°
Answer: Triangle PTQ is a right-angled isosceles triangle with the right angle at T.
Correct answer: A right-angled isosceles triangle with the right angle at T
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