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# Geometric Mean

## Key Concepts

• Understanding Ratio
• Understanding Proportion
• Understanding Geometric Mean

### Geometric Mean

Three non-zero quantities of the same kind and in the same unit are said to be continued proportion if the ratio of the first and second is the same as the ratio of second and third.

If a, b and c are in the continued proportion then a:b = b:c or b is called mean proportion.

The geometric mean of two positive numbers, a and b, is the positive number x that satisfies 𝐚/𝐱 = x/b So x2 = ab and x = √𝐚𝐛. Geometric mean is the mean proportion.

• Find the geometric mean of two numbers 4 and 25.

x = √ab

x =√4×25

=√100

= 10

• Find the geometric mean of two numbers 25 and 16.

x = √25×16

= √400

= 20

• Find the value of x, y and z.

In similar figures, corresponding sides are proportional.

Triangle BDC and Triangle ADB are similar.

x/6 = 24/x

x2   = 24 × 6

x = √24×6

= 12

Triangle BDC and Triangle ABC are similar.

6/y = y/30

y × y = 30 × 6

y2 = 30 × 6

y = √180

= 6√5

Triangle ADC and Triangle ABC are similar.

z/30 = 24/z

z × z = 30 × 24

z2 = 720

z =√720

= 12√5

x = 12, y = 6√5, z =12√5

## Exercise

• The measure of angles of triangle ABC are in the ratio 2:3:4. Find the measure of the angles.
• Solve the proportion 3/x +1 = 2/x
• If triangle ABC ~ triangle PQR, then write the corresponding angles and sides.
• Write the extremes and means of the following expressions:
1. 3:5 = 9:15
2. 2: 3 = 10:15
• Name three similar triangles. Write the ratio of their corresponding sides.
• Find the geometric mean for the following numbers:
1. 20 and 25
2. 8 and 18
• 15 is the geometric mean of 25, and what other number?
• Find the missing variable
• Find the altitude of the triangle.

### What have we learned

What is a ratio?

• Ratio is a way to compare two quantities of the same type and units.

What is proportion?

• An expression that states two ratios are equal is called proportion.

What is geometric mean?

• The geometric mean of two positive numbers, a and b, is the positive numberx that satisfies a/x = x/b . So, x2= ab and x= √ab.

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