Solving Right Triangles
A right triangle has a right angle (90º) and two acute angles.
Hypotenuse
The hypotenuse is the side opposite the right angle, and is the largest side of the triangle.
Legs or Catheti
The legs or catheti (singular: cathetus) are the sides opposite the acute angles and are the shorter sides of the triangle.
Pythagorean Theorem
In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.
Area of a Right Triangle
The area of a right triangle is equal to half the product of the legs.
Solving Right Triangles
To solve a triangle, we must find the value of all the sides and angles.
To solve a right triangle, we use trigonometric ratios and the Pythagorean theorem.
1. The Hypotenuse and a Leg Are Known
Example
Solve a right triangle knowing:
a = 415 m and b = 280 m.
sin B = 280/415 = 0.6747 B = arcsin 0.6747 = 42° 25′
C = 90° - 42° 25′ = 47° 35′
c = a cos B c = 415 · 0.7381 = 306. 31 m
2. Two Legs Are Known
Example
Solve a right triangle knowing:
b = 33 m and c = 21 m.
tan B = 33/21 = 1.5714 B = 57° 32′
C = 90° - 57° 32′ = 32° 28′
a = b/sin B a = 33/0.8347 = 39.12 m
3. The Hypotenuse and an Acute Angle Are Known
Example
Solve a right triangle knowing:
a = 45 m y B = 22°.
C = 90° − 22° = 68°
b = a sin 22° b = 45 · 0.3746 = 16.85 m
c = a cos 22° c = 45 · 0.9272 = 41.72 m
4. A Leg and an Acute Angle Are Known
Example
Solve a right triangle knowing:
b = 5.2 m and B = 37º
C = 90° − 37° = 53º
a = b/sin B a = 5.2/0.6018 = 8.64 m
c = b · cot B c = 5.2 · 1.3270 = 6. 9 m
