Solving Right Triangles

A right triangle has a right angle (90º) and two acute angles.

Right Triangle

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.

Right Triangle

Pythagorean Theorem

Pythagorean Theorem

Pythagorean Theorem


Area of a Right Triangle

The area of a right triangle is equal to half the product of the legs.

Right TriangleArea of a Triangle

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


Solving Right Triangles

Solving Right Triangles

Solving Right Triangles

Right Triangle

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


Solving Right Triangles

Solving Right Triangles

Solving Right Triangles

Triángulo

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


Solving Right Triangles

Solving Right Triangles

Solving Right Triangles

Right Triangle

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


Solving Right Triangles

Solving Right Triangles

Solving Right Triangles

Right Triangle

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