https://file.aiquickdraw.com/mathbot/file/172614865718857ohkauw.jpg

To calculate the length of \(AC\), we first need to analyze the given information in the diagram. 

Here's what we know from the image:
1. \(AB\) is perpendicular to \(BC\).
2. \(BD\) bisects the angle at point \(D\) with \(\angle DAB = 53^\circ\) and \(\angle DBC = 67^\circ\).
3. \(BC = 91 \, \text{m}\).

We need to find the length of \(AC\). 

### Steps to Solve:
1. **Identify triangles involved:**
   The triangles involved are \( \triangle ABD \) and \( \triangle BCD \).

2. **Use trigonometric ratios:**
   Since \( \angle DAB = 53^\circ \) and \( \angle DBC = 67^\circ \), and considering that \( BD \) is a straight line bisecting \( D \), we can use the trigonometric ratios to find \( AB \) and \( BC \).

3. **Calculate \(AB\) using trigonometry:**
   In \( \triangle BCD \):
   \[
   \tan(67^\circ) = \frac{BD}{BC}
   \]
   So,
   \[
   BD = BC \cdot \tan(67^\circ)
   \]
   Since \(BC = 91 \, \text{m}\),
   \[
   BD = 91 \cdot \tan(67^\circ)
   \]

4. **Calculate \(AC\):**
   To find \(AC\), we can use the cosine law or extend further using trigonometry. However, once we have \(BD\) and \(BC\) the lengths of sides complete required remain.

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Solve the geometry problem involving triangle ABC. The length of AC is calculated using trigonometric ratios and cosine law, with angles 53° and 67°, and side BC of 91m. Suitable for high school students in Grades 10-12.

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