`DeltaDEF` is right angled at E. if `tanD=12//5`, then what is the value of secF?

Options

A. `5//12`
B. `13//5`
C. `5//13`
D. `13//12`

Answer

To solve the problem, we need to find the value of secant of angle F in triangle DEF, which is right-angled at E. We are given that tan D = 12/5. Let's break down the solution step by step.

### Step 1: Understand the Triangle
We have a right triangle DEF with the right angle at E. We denote:
- DE as the base
- EF as the height (perpendicular)
- DF as the hypotenuse

### Step 2: Use the Tangent Function
From the problem, we know that:
\[ \tan D = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{EF}{DE} = \frac{12}{5} \]
This means:
- EF (opposite to angle D) = 12
- DE (adjacent to angle D) = 5

### Step 3: Calculate the Hypotenuse DF
Using the Pythagorean theorem:
\[ DF = \sqrt{DE^2 + EF^2} \]
Substituting the values:
\[ DF = \sqrt{5^2 + 12^2} \]
\[ DF = \sqrt{25 + 144} \]
\[ DF = \sqrt{169} \]
\[ DF = 13 \]

### Step 4: Find Cosine of Angle F
To find secant of angle F, we first need to find cosine of angle F. Recall that:
\[ \cos F = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
In the context of angle F:
- The adjacent side is DE (5)
- The hypotenuse is DF (13)

Thus:
\[ \cos F = \frac{DE}{DF} = \frac{5}{13} \]

### Step 5: Calculate Secant of Angle F
Secant is the reciprocal of cosine:
\[ \sec F = \frac{1}{\cos F} = \frac{1}{\frac{5}{13}} = \frac{13}{5} \]

### Final Answer
The value of secant F is:
\[ \sec F = \frac{13}{5} \]

---