Given: right-angled triangle ABC, CD-height. AD = 24cm, DB = 54cm, angle C = 90 degrees. Find AC, BC

Let us denote the length of the height CD x.
Consider a right-angled triangle ADC and write down the Pythagorean theorem in it.
AC² = CD² + AD² = x² + 576;
Do the same for triangle CDB.
BC² = CD² + DB² = x² + 2916.
In triangle ABC we write down the square of the hypotenuse AB.
AB² = AC² + BC² = (AD + DB) ² = (24 + 54) ² = 78² = 6084;
x² + 576 + x² + 2916 = 6084
2x² = 2592
x² = 1296
x = 36.
AC = √ (x² + 576) = √ (1296 + 576) = √1872 = 12√13 (cm).
BC = √ (x² + 2916) = √ (1296 + 2916) = √4212 = 18√13 (cm).
Answer: AC = 12√13 cm, BC = 18√13 cm.



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