Triangle Solving Solutions
30 curated triangle solving problems covering sine law, cosine law, and core problem-solving patterns with full derivations.
30 problems
Problem 1: In an acute , suppose that and
In an acute , suppose that and . Which of the following values can take? A. 5 B. 4 C. D. 3
Problem 2: Determine angle
In , let the sides opposite be , respectively. Suppose (1) Determine angle . Let be the midpoint of . (2) If and the area of is , find .
Problem 3: Multiple choice (select all that apply)
Multiple choice (select all that apply). In , let the sides opposite be . Given , which statements are true? A. B. is obtuse C. If the circumradius is and the i
Problem 4: Multiple choice (select all that apply)
Multiple choice (select all that apply). In , let the sides opposite be . Suppose . Which statements are true? A. B. If and , then C. If , then the maximum poss
Problem 5: determine the range of
In a plane quadrilateral , suppose , , and . When the length of the diagonal is minimized, determine the range of .
Problem 6: Find the range of Which option is correct
In an acute , let the sides opposite be . Suppose . Find the range of Which option is correct? A. B. C. D.
Problem 7: find
In , bisects and meets at ; bisects and meets at . Given and , find .
Problem 8: Compute
A point inside is called a Brocard point if ; is the Brocard angle. It satisfies Compute . A. B. C. D.
Problem 9: Determine angle
In , let the sides opposite be , respectively, and suppose (1) Determine angle . (2) If the area of is and the perimeter is , find .
Problem 10: Find the range of
An acute triangle is inscribed in the unit circle (so its circumradius is ). Let the sides opposite be . Suppose Find the range of . A. B. C. D.
Problem 11: Find the maximum possible value of the perimeter of
In , let the sides opposite be . Suppose Find the maximum possible value of the perimeter of . A. B. C. 6 D. 9
Problem 12: Find the range of
In , let the sides opposite be . Suppose and . Find the range of .
Problem 13: Find the radius of circle
Ptolemy's theorem states that for a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides. Let be a cyclic quadril
Problem 14: Find the range of
In , let the sides opposite be . The circumradius is , and Find the range of . A. B. C. D.
Problem 15: Find angle
In , let the sides opposite be . Suppose (1) Find angle . (2) If the area of is , find the minimum possible value of , and determine the triangle's shape at equ
Problem 16: In , consider the statement: form an arithmetic progression and form a geometric progression
In , consider the statement: form an arithmetic progression and form a geometric progression. This statement is ( ) of: is equilateral. A. Sufficient but not ne
Problem 17: Multiple choice (select all that apply)
Multiple choice (select all that apply). Which of the following statements are true? A. In , if then B. In an acute , the inequality always holds C. In , if , t
Problem 18: Find the range of
In an acute , let the sides opposite be . Suppose . Find the range of
Problem 19: Find the area of
In , the circumradius is , and Find the area of . A. B. C. D.
Problem 20: In , let the sides opposite be
In , let the sides opposite be . Suppose and . Which statement is correct? A. is right-angled B. is acute C. is obtuse D. The type of cannot be determined
Problem 21: find the maximum possible value of
Let and be unit vectors with . If a vector satisfies , find the maximum possible value of .
Problem 22: find angle
In , let the sides opposite be . Given find angle . A. B. C. D.
Problem 23: Determine the logical relationship between the statements: (Condition) (Conclusion) A
In , let the sides opposite be . Determine the logical relationship between the statements: (Condition) (Conclusion) A. Sufficient but not necessary B. Necessar
Problem 24: find
In , let the sides opposite be . If find . A. B. C. D.
Problem 25: Find the sine of the smallest angle
In a right triangle, the cosines of its three interior angles form an arithmetic progression. Find the sine of the smallest angle. A. B. C. D.
Problem 26: Find
In , and . Find .
Problem 27: Find the range of
In , let the sides opposite be . Suppose . Find the range of .
Problem 28: compute
In , if , compute
Problem 29: find
In , let the sides opposite be . Given , , and , find .
Problem 30: The Zhenguo Temple Pagoda is a seven-story brick tower
The Zhenguo Temple Pagoda is a seven-story brick tower. A student wants to estimate its height . A building of height lies due north of the pagoda. From a point