Question

determine the following. *

1. in the diagram below, what is ac?

diagram of a triangle with right angles at b and d, segments labeled: ab (vertical), bc (horizontal), ed (vertical segment from d to e on ac), ae = 8, ec = x + 11, bd = 7, dc = x + 9

Explanation:

Answer:

To determine the length of \( AC \), we use the Basic Proportionality Theorem (Thales' theorem), which states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. Here, \( ED \parallel AB \), so:

Step 1: Set up the proportion

Since \( ED \parallel AB \), we have \(\frac{AE}{EC}=\frac{BD}{DC}\). Substituting the given values:

$$ \frac{8}{x + 11}=\frac{7}{x + 9} $$

Step 2: Cross - multiply

Cross - multiplying gives us:

$$ 8(x + 9)=7(x + 11) $$

Step 3: Expand both sides

Expanding the left - hand side: \(8x+72\), and the right - hand side: \(7x + 77\). So the equation becomes:

$$ 8x+72 = 7x+77 $$

Step 4: Solve for \( x \)

Subtract \(7x\) from both sides: \(8x - 7x+72=7x - 7x + 77\), which simplifies to \(x + 72=77\). Then subtract 72 from both sides: \(x=77 - 72=5\)

Step 5: Find the length of \( AC \)

The length of \( AC=AE + EC\). We know that \( AE = 8\) and \( EC=x + 11\). Substitute \(x = 5\) into \(EC\): \(EC=5 + 11 = 16\). Then \(AC=8 + 16=24\)

So the length of \( AC \) is \(\boldsymbol{24}\)