The height of a right-angled triangle, drawn from the vertex of the right angle to the hypotenuse, is 4√3. One of the legs is 8. Find the area of the original triangle.
1. A, B, C – the vertices of the triangle. ∠С = 90 °. Height CE = 4√3 units. BC = 8 units.
2. BE = √BC² – CE² = √8² – (4√3) ² = √64 – 48 = √16 = 4 units.
3. We calculate the length of the segment AE, applying the formula for calculating the length of the height CE, drawn from the vertex of an angle equal to 90 °:
CE = √AE x BE.
CE² = AE x BE. AE = CE²: BE = 48: 4 = 12 units.
4. AB = AE + BE = 12 + 4 = 16 units.
5. Area of the triangle = AB x CE / 2 = 16 x 4√3 / 2 = 32√3 units.
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