The problem shows a triangle $\triangle ABC$ with a segment $DE$ drawn parallel to $AB$. From the markings, the given lengths are:

• $DE = 4x - 5$
• $AB = 7x + 4$

Solving:

Since $DE \parallel AB$, the triangles $\triangle CDE$ and $\triangle CAB$ are similar by the Basic Proportionality Theorem (or Thales' theorem). Therefore, the ratio of the sides of the smaller triangle is equal to the ratio of the sides of the larger triangle:

$\frac{DE}{AB} = \frac{CD}{CB}$

From the diagram, $CD = CB$, as indicated by the tick marks. This implies $\frac{CD}{CB} = \frac{1}{2}$.

Thus, we can write:

$\frac{4x - 5}{7x + 4} = \frac{1}{2}$

Cross-multiply to eliminate the fraction:

$2(4x - 5) = 7x + 4$

Simplify:

$8x - 10 = 7x + 4$

Solve for $x$:

$x = 14$

Finding $DE$ and $AB$:

1. Substitute $x = 14$ into $DE = 4x - 5$: $DE = 4(14) - 5 = 56 - 5 = 51$

2. Substitute $x = 14$ into $AB = 7x + 4$: $AB = 7(14) + 4 = 98 + 4 = 102$

Final Answers:

• $x = 14$
• $DE = 51$
• $AB = 102$

Would you like a detailed explanation of the proportionality theorem used here or additional practice questions?

Related Questions:

1. What is the significance of parallel lines in similar triangles?
2. How does the Basic Proportionality Theorem work?
3. What other methods could be used to confirm the values of $DE$ and $AB$?
4. Can this theorem be applied in 3D geometry?
5. What happens if $DE$ is not parallel to $AB$?

Tip:

When dealing with similar triangles, always verify which sides or angles correspond to ensure the correct ratio setup!