Question

In ΔOPQ, the measure of ∠Q=90°, QP = 11, OQ = 60, and PO = 61. What ratio represents the tangent of ∠O?

Answer 1

In triangle OPQ with a right angle at ∠Q, the tangent of ∠O is 61:60, calculated as the ratio of the side opposite (∠O) to the adjacent side (OQ).

In triangle OPQ, with a right angle at ∠Q, the tangent of an angle (∠O in this case) is the ratio of the length of the side opposite the angle to the length of the adjacent side. Using the given information, ∠Q=90°, QP=11, OQ=60, and PO=61, we can calculate the tangent of ∠O.

The tangent (\(\tan\)) of ∠O is expressed as:

`\[ \tan(\angle O) = \frac{\text{Opposite Side}}{\text{Adjacent Side}} \]`

In this scenario, \(OP\) is the side opposite ∠O, and \(OQ\) is the adjacent side. Therefore:

`\[ \tan(\angle O) = (OP)/(OQ) \]`

Substituting the given values, where \(OP = 61\) and \(OQ = 60\), we get:

`\[ \tan(\angle O) = (61)/(60) \]`

So, the ratio representing the tangent of ∠O is \(61:60\). This means that for every 61 units along the side opposite ∠O, there are 60 units along the adjacent side in the right-angled triangle OPQ. This ratio characterizes the relationship between the lengths of these sides and defines the tangent of ∠O in the given triangle.