In triangle DEF, angle E = 90 degrees. Find EF if DE = 6 cm and tg F = 0.5.

Okay, let's solve this geometry problem step-by-step!

We are given a right-angled triangle △DEF, where the angle ∠E is 90°. We know that the length of side DE is 6 cm, and the tangent of angle F (tg F) is 0.5.

We need to find the length of side EF.

Here's what we know:

• ∠E = 90°
• DE = 6 cm
• tg F = 0.5

Recall the definition of the tangent of an angle in a right-angled triangle:

tg(angle) = (opposite side) / (adjacent side)

In our triangle △DEF, with respect to angle F:

• The opposite side is DE.
• The adjacent side is EF.

So, we can write the formula for tg F as:

$$ \text{tg } F = \frac{DE}{EF} $$

Now, let's plug in the values we know:

$$ 0.5 = \frac{6 \text{ cm}}{EF} $$

To find EF, we can rearrange the equation:

$$ EF = \frac{6 \text{ cm}}{0.5} $$

Calculating this gives us:

$$ EF = 12 \text{ cm} $$

Check: If EF = 12 cm and DE = 6 cm, then tg F = 6/12 = 0.5, which matches the given information.

Therefore, the length of side EF is 12 cm.

Answer: 12 cm