Question

find the secant of ∠d.

simplify your answer and write it as a proper fraction, improper fraction, or whole number.

sec(d) =

submit

Explanation:

Step1: Recall the definition of secant

The secant of an angle in a right triangle is the reciprocal of the cosine of the angle. For an angle \( \theta \), \( \sec(\theta) = \frac{1}{\cos(\theta)} \), and \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \), so \( \sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} \).

Step2: Identify the sides relative to \( \angle D \)

In right triangle \( DCE \) (right-angled at \( C \)), for \( \angle D \):

• The hypotenuse is \( DE = 26 \).
• First, we need to find the length of the adjacent side to \( \angle D \). The opposite side to \( \angle D \) is \( CE = 24 \). Using the Pythagorean theorem \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse, and let the adjacent side ( \( DC \)) be \( x \). So \( x^2 + 24^2 = 26^2 \).
• Calculate \( x^2 \): \( x^2 = 26^2 - 24^2 = (26 - 24)(26 + 24) = 2 \times 50 = 100 \), so \( x = 10 \) (since length is positive). So the adjacent side to \( \angle D \) is \( DC = 10 \), and the hypotenuse is \( DE = 26 \).

Step3: Calculate \( \sec(D) \)

Using the definition of secant, \( \sec(D) = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{DE}{DC} = \frac{26}{10} \). Simplify this fraction by dividing numerator and denominator by 2: \( \frac{26 \div 2}{10 \div 2} = \frac{13}{5} \).

Answer:

\( \frac{13}{5} \)