Question

which one of the following sets of data determines a unique triangle? choose the correct answer below a. a = 7, b = 24, c = 25 b. a = 50°, b = 50°, c = 80° c. a = 3, b = 5, c = 20 d. a = 40°, b = 20°, c = 30°

Explanation:

Step1: Check triangle inequality for option A

For a triangle with sides \(a = 7\), \(b = 24\), \(c = 25\).
Check \(a + b>c\): \(7+24 = 31>25\)
Check \(a + c>b\): \(7 + 25=32>24\)
Check \(b + c>a\): \(24+25 = 49>7\)
Also, since the side - lengths are fixed, by SSS (Side - Side - Side) congruence criterion, a unique triangle is formed.

Step2: Analyze option B

For angles \(A = 50^{\circ}\), \(B = 50^{\circ}\), \(C=80^{\circ}\).
By AAA (Angle - Angle - Angle) similarity criterion, infinitely many similar (but not necessarily congruent) triangles can be formed.

Step3: Check triangle inequality for option C

For sides \(a = 3\), \(b = 5\), \(c = 20\)
\(a + b=3 + 5=8<20\). So, no triangle can be formed.

Step4: Check sum of angles for option D

For angles \(A = 40^{\circ}\), \(B = 20^{\circ}\), \(C = 30^{\circ}\)
\(A + B+C=40^{\circ}+20^{\circ}+30^{\circ}=90^{\circ}
eq180^{\circ}\). So, no triangle can be formed.

Answer:

A. \(a = 7\), \(b = 24\), \(c = 25\)