Question

Use the proportion \(\frac{a}{b}=\frac{b}{d},\) where \(a\) \(\boldsymbol{b},\) and \(\boldsymbol{d}\) are positive numbers. In the proportion \(\frac{a}{b}=\frac{b}{d}, b\) is called the geometric mean of \(a\) and \(d .\) Use the cross product property to show that \(b=\sqrt{a d}\).

Short answer

Based on the cross-product property, it has been shown that in the proportion \(\frac{a}{b}=\frac{b}{d}\), \(b\) equals \(\sqrt{ad}\)

Step-by-step solution

Write down the proportion

First write down the proportion: \(\frac{a}{b}=\frac{b}{d}\)

Apply the cross product property

Applying the cross-product property, which states that in the equality of two fractions, the product of the means equals the product of the extremes, gives \(a * d = b * b\), which simplifies to \(ad = b^2\).

Determine the value of \(b\)

Finally, to find the value of \(b\), take the square root of both sides: \(b = \sqrt{ad}\)