x = 32, y = 11.31, z = 33.94
Step-by-step explanation:
Step 1: Let the triangle be ΔABC with B as the right angle. Let the hypotenuse be intersected by the altitude drawn from B at D. So there are 3 triangles, ΔABC, ΔADB and ΔBDC.
Step 2: To find x, y and z, use Pythagoras theorem where square of hypotenuse is equal to sum of squares of the other two sides of the right triangle.
Step 3: Make equations based on the theorem for all 3 triangles.
In ΔABC, (4+x)² = 12² + z²
⇒ 16 + 8x + x² = 144 + z²
⇒ x² + 8x - z² = 128 --------- (1)
In ΔADB, 12² = 4² + y²
⇒ 144 = 16 + y²
⇒ y² = 128 ------- (2)
In ΔBDC, z² = x² + y²
⇒ z² = 128 + x² ------ (3)
Substituting (3) in (1) ⇒ x² + 8x - (128 + x²) = 128
⇒ x² + 8x - 128 - x² = 128
⇒ 8x = 256
⇒ x = 32
From (3), z² = 128 + 32² = 128 + 1024 = 1152
⇒ z = 33.94
From (2), y² = 128
⇒ y = 11.31
