We have a triangle $\triangle ABC$ with a perpendicular from point $D$ to side $AC$, forming two right-angled triangles, $\triangle BDA$ and $\triangle BDC$. The sides $BD$, $AD$, and $CD$ are marked as equal, indicating the triangle is isosceles.

Given the following equations for the sides:

• $AD = 3x + 7$
• $CD = 2x + 11$

Since $AD = CD$, we can set these two expressions equal to each other and solve for $x$:

$3x + 7 = 2x + 11$

Let's solve for $x$.

$3x - 2x = 11 - 7$

$x = 4$

Now that we have $x = 4$, we can find the lengths of $AD$ and $CD$ by substituting $x = 4$ into the given expressions.

For $AD$: $AD = 3(4) + 7 = 12 + 7 = 19$

For $CD$: $CD = 2(4) + 11 = 8 + 11 = 19$

Thus:

• $x = 4$
• $AD = 19$
• $CD = 19$

If you need further clarification or details, feel free to ask.

Here are 5 related questions to expand your understanding:

1. How would the problem change if the sides were not equal?
2. How can we use the Pythagorean Theorem to calculate $BD$?
3. Can we use trigonometric ratios to find angles in this triangle?
4. How would the solution differ if $AD \neq CD$?
5. What is the significance of $D$ being a right angle in this context?

Tip: Always look for symmetry in geometric problems, as it can simplify calculations significantly!