Solve Triangle: a=5, b=12, c=13
Complete solution for the triangle with given measurements. Find all missing sides, angles, area, and perimeter.
Area: 30 sq units
Perimeter: 30 units
Step-by-Step Solution
Step 1
Given Information
Sides: a = 5, b = 12, c = 13
Angles: A = ?°, B = ?°, C = ?°
Step 2
Method: SSS (Side-Side-Side)
All three sides are known
Use Law of Cosines to find angles
Step 3
Find Angles (Law of Cosines)
cos(A) = (b² + c² - a²) / (2bc) = 0.9231
A = 22.62°
cos(B) = (a² + c² - b²) / (2ac) = 0.3846
B = 67.38°
C = 180° - A - B = 90°
Step 4
Calculate Area (Heron's Formula)
s = (a + b + c) / 2 = 15
Area = √[s(s-a)(s-b)(s-c)]
Area = √[15 × 10 × 3 × 2]
Area = 30
Step 5
Perimeter
P = a + b + c = 30
Step 6
Triangle Classification
Scalene (no sides equal)
Right triangle (one 90° angle)
Step 7
Final Answer
A = 22.62°, B = 67.38°, C = 90°, Area = 30
Triangle Solution Methods
SSS (3 sides): Use Law of Cosines for angles, Heron's for area
SAS (2 sides, included angle): Use Law of Cosines
ASA/AAS (2 angles, 1 side): Use Law of Sines