Question

Solve for the missing variable.

Answer 1

Answer:

• x ≈ 6.3
• y ≈ 7.7
• z ≈ 4.9

Step-by-step explanation:

In this geometry, there are three "geometric mean" relations that apply. They can be summarized by the rule ...

  a segment touching the hypotenuse is equal to the geometric mean of the segments of the hypotenuse it touches.

For x and y, the two segments will be the short one (4 or 6, respectively) and the entire hypotenuse (4+6=10).

Using this rule, we have ...

  x = √(4·10) ≈ 6.3

  y = √(6·10) ≈ 7.7

  z = √(4·6) ≈ 4.9

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The geometric mean of two numbers is the square root of their product.

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Additional comment

These relations come about because all of the right triangles in this figure are similar. That means corresponding sides are proportional.

  x/10 = 4/x   ⇒   x² = 40

  y/10 = 6/y   ⇒   y² = 60

  z/6 = 4/z   ⇒   z² = 24