1. **Problem statement:** Find the height of triangle ABC where sides AB and BC are 10 units each, base AC is 8 units, and a perpendicular from B to AC divides AC into two equal parts of 4 units each. 2. **Formula used:** To find the height (h), we use the Pythagorean theorem in the right triangle formed by the height, half the base, and the side AB (or BC). 3. **Applying Pythagorean theorem:** $$h = \sqrt{AB^2 - (\frac{AC}{2})^2}$$ 4. **Substitute values:** $$h = \sqrt{10^2 - 4^2} = \sqrt{100 - 16} = \sqrt{84}$$ 5. **Calculate height:** $$h = \sqrt{84} \approx 9.165$$ 6. **Explanation:** The height is the perpendicular segment from B to AC, which splits AC into two equal parts of 4 units each. Using the Pythagorean theorem, we find the height by subtracting the square of half the base from the square of the side length. 7. **Regarding your results:** - You got $\sqrt{86}$ which is close but slightly off; the correct value is $\sqrt{84}$. - Using trigonometric functions like sine, cosine, or tangent should give approximately the same height $9.165$. **Final answer:** The height of the triangle is approximately $9.165$ units.