Reference the right triangle shown below to write the quantity as a fraction in lowest terms

tan P

A right angled triangle is a triangle where one of the internal angles is 90°.

Consider the general right angle triangle, in figure 1 below, with angle β and sides labeled as shown below

Right-Angle-Triangle

The sides are labeled with respect to the angle β, so that the nearest side is called the adjacent side, the side that is directly opposite is called the opposite side while the longest side is called the hypotenuse.

Therefore, given any angle in a triangle, it should be easy to identify the adjacent, the opposite side and the hypotenuse.

Example

Reference to the figure below.

Example1

1. Identify the right angle triangles in the above diagram.

Solution: A right angled triangle is a triangle where one of the internal angles is 90°. Such triangles are ABC, ABN, DAC and DNC.

2. Identify the adjacent sides to angles θ, α, β and µ.

Solution:

Angle θ is in triangle DAC. The angle is nearer to line AC hence, it is the adjacent line.

Angle α is in triangle ABN. The angle is nearer to line AB hence, it is the adjacent line.

Angle β is in triangle DNC. The angle is nearer to line DC hence, it is the adjacent line.

Angle µ is in triangle ABC. The angle is nearer to line AB hence, it is the adjacent line.

3. Identify the opposite sides to angles θ, α, β and µ.

Solution:

Angle θ is in triangle DAC. The angle is opposite to line DC hence, it is the opposite line.

Angle α is in triangle ABN. The angle is opposite to line AN hence, it is the opposite line.

Angle β is in triangle DNC. The angle is opposite to line CN hence, it is the opposite line.

Angle µ is in triangle ABC. The angle is opposite to line AC hence, it is the opposite line.

4. Identify hypotenuse with respect to angles θ, α, β and µ.

Solution:

Angle θ is in triangle DAC. The longest side in that angle triangle is AD hence, it is the hypotenuse.

Angle α is in triangle ABN. The longest side in that angle triangle is NB hence, it is the hypotenuse.

Angle β is in triangle DNC. The longest side in that angle triangle is DN hence, it is the hypotenuse.

Angle µ is in triangle ABC. The longest side in that angle triangle is CB hence, it is the hypotenuse.

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