Question

Find x, y, and z. For number 24

Answer 1

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