University of the Philippines College Admission Test · Mathematics

Euclidean Geometry and Trigonometry

35 practice questions, each with the answer and an explanation.

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

All 35 questions

  1. The interior angles of a triangle are in the ratio 2 : 3 : 5. What is the measure of the largest angle?
  2. A right triangle has legs of lengths 9 cm and 12 cm. What is the length of its hypotenuse?
  3. What is the area of a circular sector with radius 6 cm and central angle 60 degrees?
  4. An inscribed angle in a circle intercepts an arc measuring 110 degrees. What is the measure of the inscribed angle?
  5. Two similar triangles have corresponding sides in the ratio 3 : 5. If the area of the smaller triangle is 36 cm^2, what is the area of the larger triangle?
  6. What is the distance between the points (-2, 5) and (4, -3) in the Cartesian plane?
  7. What are the coordinates of the midpoint of the line segment joining A(3, -7) and B(-5, 1)?
  8. What is the standard equation of the line perpendicular to 2x - 3y = 6 and passing through (4, 1)?
  9. What is the expanded equation of a circle centered at (2, -3) with a radius of 5 units?
  10. Evaluate the exact numerical expression: sin^2(25 deg) + cos^2(25 deg) + tan(45 deg).
  11. What is the exact value of sin(150 degrees)?
  12. Find all solutions to 2 sin(x) - 1 = 0 on the interval [0, 2pi].
  13. In triangle ABC, side a = 5 cm, side b = 7 cm, and included angle C = 60 degrees. What is the length of side c?
  14. In triangle ABC, angle A = 30 degrees, side a = 8 cm, and angle B = 45 degrees. What is the length of side b?
  15. If sin(x) = 3/5 with x in the first quadrant, what is the exact value of sin(2x)?
  16. What is the volume of a sphere with a radius of 3 cm?
  17. A right circular cone has a base radius of 5 cm and a height of 12 cm. What is its lateral surface area?
  18. What is the area of a trapezoid with parallel bases 14 cm and 22 cm, and an altitude of 8 cm?
  19. What is the area of a regular hexagon with a side length of 6 cm?
  20. The diagonals of a rhombus are 12 cm and 16 cm. What is its perimeter?
  21. A point P is 13 cm from the center of a circle of radius 5 cm. What is the length of the tangent line segment from P to the circle?
  22. From a point on flat ground 40 meters from the base of a vertical tower, the angle of elevation to the top is 30 degrees. What is the height of the tower?
  23. A parallelogram has adjacent sides of lengths 8 cm and 10 cm, enclosing an angle of 30 degrees. What is its area?
  24. What is the volume of a right triangular prism whose base is a right triangle with legs 6 cm and 8 cm, and whose height is 15 cm?
  25. A chord in a circle of radius 10 cm is located 6 cm from the center. What is the total length of the chord?
  26. Two chords AB and CD intersect at P inside a circle. If AP = 4 cm, PB = 6 cm, and CP = 3 cm, what is the length of PD?
  27. From an external point P, a tangent PT of length 6 cm and a secant PAB are drawn such that PA = 4 cm. What is the length of chord AB?
  28. What are the coordinates of the centroid of a triangle with vertices at (1, 4), (5, 8), and (6, -3)?
  29. For the ellipse x^2 / 25 + y^2 / 16 = 1, what are the coordinates of its foci?
  30. What is the arc length intercepted by a central angle of 72 degrees in a circle with a radius of 10 cm?
  31. What is the sum of the interior angles of a convex decagon (10-sided polygon)?
  32. What is the measure of each exterior angle of a regular dodecagon (12-sided polygon)?
  33. Simplify the trigonometric expression: (sin(x) / (1 + cos(x))) + ((1 + cos(x)) / sin(x)).
  34. Using Heron's formula, what is the area of a triangle with side lengths 7 cm, 8 cm, and 9 cm?
  35. What is the perpendicular distance from the point (2, 3) to the straight line 3x + 4y - 8 = 0?

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