Q1. What is the length of the radius of a circle if the circumference is 31.4 cm?
Solution:
The circumference C of a circle is given by C = 2πr. Thus, r = C / (2π) = 31.4 / (2π) ≈ 5 cm.
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Q2. Two triangles are similar if their corresponding angles are equal. If triangle DEF is similar to triangle XYZ, and angle D = 30 degrees, what is the measure of angle X?
Solution:
Since the triangles are similar, corresponding angles are equal. Therefore, angle X = angle D = 30 degrees.
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Q3. If the diameter of a circle is 14 cm, what is the circumference?
Solution:
The circumference C of a circle is given by C = πd. Thus, C = π(14) ≈ 44 cm.
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Q4. What is the length of the tangent from a point P outside a circle to the point of tangency T if the radius of the circle is 5 cm and the distance from P to the center O of the circle is 13 cm?
Solution:
Using the tangent-secant theorem, PT² = PO² - OT². Thus, PT² = 13² - 5² = 169 - 25 = 144, so PT = 12 cm.
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Q5. In triangle PQR, if PQ = 8 cm, PR = 6 cm, and QR = 10 cm, is triangle PQR a right triangle?
Solution:
Using the Pythagorean theorem, if PQ² + PR² = QR², then it is a right triangle. 8² + 6² = 64 + 36 = 100 = 10².
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Q6. If triangle DEF is similar to triangle XYZ, and the lengths of DE and XY are 4 cm and 8 cm respectively, what is the ratio of their areas?
Solution:
The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides. Thus, (4/8)² = 1/4.
