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A triangle was dilated by a scale factor of 2. If cos a° = three fifths and segment FD measures 6 units, how long is segment DE?

triangle DEF in which angle F is a right angle, angle D measures a degrees, and angle E measures b degrees

segment DE = 3.6 units
segment DE = 8 units
segment DE = 10 units
segment DE = 12.4 units

1 answer

To solve for segment DE in triangle DEF, we note a few important properties of the triangle.

  1. Understanding the angles: Since angle F is a right angle and \( \angle D = a^\circ \), we can express the cosine of angle D (a) using the adjacent and hypotenuse relationship in triangle DEF: \[ \cos a = \frac{\text{adjacent (side DF)}}{\text{hypotenuse (side DE)}} \]

  2. Given that \( \cos a = \frac{3}{5} \), we will set:

    • Side DF as the side adjacent to angle D
    • Side DE as the hypotenuse

    Therefore: \[ \frac{DF}{DE} = \frac{3}{5} \]

  3. Finding side DE: Using the information provided:

    • Segment FD measures 6 units, which we assume corresponds to the adjacent side DF in this context.
    • Since we need to find DE (the hypotenuse): \[ DF = 6 \]

    Substitute DF and solve for DE: \[ \frac{6}{DE} = \frac{3}{5} \]

    Cross-multiply and solve for DE: \[ 3 \cdot DE = 6 \cdot 5 \] \[ 3 \cdot DE = 30 \] \[ DE = \frac{30}{3} = 10 \text{ units} \]

  4. Factor of dilation: The problem states that the triangle was dilated by a scale factor of 2. Therefore, we need to multiply our previously calculated DE by this scale factor: \[ DE' = 10 \times 2 = 20 \text{ units} \]

However, the problem asks us about possible values of segment DE in the undeformed triangle:

  1. The options provided are:
    • 3.6 units
    • 8 units
    • 10 units
    • 12.4 units

Since we calculated that segment DE in the original triangle measures 10 units, that corresponds with the option listed.

Thus, the answer is:

Segment DE = 10 units.