Answer:
x = 4
Explanation:
Similarity of triangle ABC and DBE:
- Triangles ABC and DBE are similar by the Angle-Angle (AA) similarity rule.
- Since DE is parallel to AC, we have two sets of corresponding angles, which are equal.
- Angles BDE and BAC are the first set of corresponding angles, while angles BED and BCA are the other set.
- In similar triangles, the sides are also similar and proportional.
- This means that sides BD and BA are similar and proportional (BD / BA), while sides BE and BC are similar and proportional (BE / BC).
Now we can use these proportions to find x:
BD / BA = BE / BC
(x + 2) / (x + 2 + x) = 3 / (3 + 2)
Combining like terms gives us:
(x + 2) / 2x + 2 = 3 / 5
Cross multiplying gives us:
3(2x + 2) = 5(x + 2)
6x + 6 = 5x + 10
Subtracting 6 from both sides gives us:
(6x + 6 = 5x + 10) - 6
6x = 5x + 4
Subtracting 5x from both sides gives us:
(6x = 5x + 4) - 5x
x = 4
Thus, x = 4
Check the validity of the answer:
We can check our answer by plugging in 4 for x and seeing if we get 3/5 on both sides:
(4 + 2) / (4 + 2 + 4) = 3/5
6 / 10 = 3/5
3/5 = 3/5
Thus, our answer is correct.