How to Find a Mean Proportion?

Question 4 Exercise 6(B)

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Answer:

Solution:

(i) mean proportional between 16 and 4

=\sqrt{16\times4}

=\sqrt{64}

= 8

(ii) mean proportional between 3 and 27

=\sqrt{3\times27}

=\sqrt{81}

= 9

(iii) mean proportional between 0.9 and 2.5

=\sqrt{0.9\times2.5}

=\sqrt{\frac{9}{10}}\times\sqrt{\frac{25}{10}}

=\sqrt{\frac{225}{100}}

=\frac{15}{10}

= 1.5

(iv) mean proportional between 0.6 and 9.6

=\sqrt{0.6\times9.6}

=\sqrt{\frac{6}{10}}\times\sqrt{\frac{96}{10}}

=\sqrt{\frac{576}{100}}

=\frac{24}{10}=2.4

(v) mean proportional between \frac{1}{4}\ and\ \frac{1}{16}

=\sqrt{\frac{1}{4}\times\frac{1}{16}}

=\sqrt{\frac{1}{64}}

=\frac{1}{8}

Video transcript

"Hello, Welcome to Lido My name is Nestor. We're going to see find the main properties for the given equation. So here we have my fish centers. We should going to find a means of motion for the given for the first 116. And for the finder me proportion are it is the fax number is going to be root of a to be 2016 into for the answer is wait a second one if you find the same way three into 27 2003 in. Going to be root of 81. It means nice. Okay, but R1 Rudolph 0.9 into 2.5. It's going to be Line by 25 2009 by 10 and into root over 25 by 10 C equals to root over 225 by hundred the means 1.56. Imagine. Okay, the fourth one 0.63 6 by 10 to power 6 by 10 into root over there is nothing 9610 5 so 9 is 6 into 6 is going to be my 7500 value is equal to make it 4 by 10. It means 2.0. My poor brother 1 by 4 n plus 1 by 2 grey area between 4 to 1 by 2. It is 1 by 8. I hope you understand this video subscribe this channel for weekly updates are aggressive actions, and thanks for watching this video."

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Mean proportional is another term for geometric mean. Geometric mean is a measure of central tendency, like the arithmetic mean. It is calculated differently, however. It is a common measurement in ratios and proportions.

How to Calculate Mean Proportional

Here, we are going to discuss how the mean proportional is defined, and how to calculate it.

Definition and Formula

For two real, positive numbers \(a\) and \(b\), the geometric mean (also known as mean proportional) is the number \(x\) satisfying:

\(\frac{a}{x} = \frac{x}{b}\)

In other words,

\(x  = \sqrt{ab}\)

Note that the term inside the square root must be positive for this formula to work.

Mean Proportional Between Two Numbers

As we have seen above, to calculate the mean proportional between two numbers \(a\) and \(b\) we have to:

1. Take the product of the two numbers, \(ab\)
2. Take the square root of this product.

This gives us the mean proportional between the two numbers, or the geometric mean.

Applications of Mean Proportional

Geometric Mean Altitude Theorem

In a right angled triangle ABD, if we drop a perpendicular AC onto hypotenuse BD, we have:

\(\angle ACD = \angle DAB = 90 \circ\)

As well as,

\(\angle ADB = \angle BAC\)

Thus, by AA similarity criterion,

\(\triangle ADB \sim \triangle CDA \sim \triangle CAB\)

This gives us, by definition of similarity,

\(\frac{CD}{AC} = \frac{AC}{CB}\)

In other words, AC is the geometric mean of CD and CB. 

\(AC = \sqrt{CD\cdot CB}\)

We can get CD and CB from similar relations:

\(\frac{CD}{DA} = \frac{DA}{DB}\)

Or,

\(CD = \frac{{(DA)}^2}{DB}\)

Similarly,

\(CB = \frac{{(AB)}^2}{DB}\)

Solved Examples

Question 1. Calculate the mean proportional between 234 and 104.

Solution. Here, we can write \(a\) = 234, \(b\) = 104. Then, by definition of geometric mean, 

\(x = \sqrt{ab}\) 

Substituting \(a\) and \(b\), we have,

\(x = \sqrt{234 \cdot 104}\)

\(x = \sqrt{24336} = 156\)

Thus, the geometric mean of 234 and 104 is 156.

Question 2. Calculate the length of altitude on hypotenuse, for a triangle with sides 3 cm, 4 cm and 5 cm.

Solution. Using above figure again for reference, let AB = 3 cm, AD = 4 cm, DB = 5 cm.

From Geometric Mean Altitude Theorem, we know that the altitude AC is given by,

\(AC = \sqrt{CD \cdot CB}\)

We obtain CD and CB separately as follows:

\(CD = \frac{{(DA)}^2}{DB}\)

\(CD = \frac{16}{5} = 3.2\)

And,

\(CB = \frac{{(AB)}^2}{DB}\)

\(CB = \frac{9}{5} = 1.8\)

So we now have altitude AC as,

\(AC = \sqrt {1.8 \cdot 3.2} = \sqrt{5.76} = 2.4\)

Thus, the length of the altitude is 2.4 cm.

FAQs

How do you find the mean proportional of A and B?

The mean proportional x of A and B is defined as (x = \sqrt{A \cdot B}).

What is the mean proportional of 4 and 9?

The mean proportional or geometric mean \(x\) of 4 and 9 is given by,\(x = \sqrt{4 \cdot 9} = \sqrt{36}\)\(x = 6\)

Thus, 6 is the geometric mean of 4 and 9.

We will learn how to find the mean and third proportional of the set of three numbers.

If x, y and z are in continued proportion then y is called the mean proportional (or geometric mean) of x and z.

If y is the mean proportional of x and z, y^2 = xz, i.e., y = +\(\sqrt{xz}\).

For example, the mean proportion of 4 and 16 = +\(\sqrt{4 × 16}\)  = +\(\sqrt{64}\) = 8

If x, y and z are in continued proportion then z is called the third proportional.

For example, the third proportional of 4, 8 is 16.

Solved examples on understanding mean and third proportional

1. Find the third proportional to 2.5 g and 3.5 g.

Solution:

Therefore, 2.5, 3.5 and x are in continuous proportion.

 \(\frac{2.5}{3.5}\) = \(\frac{3.5}{x}\)

⟹ 2.5x = 3.5 × 3.5

⟹ x = \(\frac{3.5 × 3.5}{2.5}\)

⟹ x = 4.9 g

2. Find the mean proportional of 3 and 27.

Solution:

The mean proportional of 3 and 27 = +\(\sqrt{3 × 27}\) = +\(\sqrt{81}\) = 9.

3. Find the mean between 6 and 0.54.

Solution:

The mean proportional of 6 and 0.54 = +\(\sqrt{6 × 0.54}\) = +\(\sqrt{3.24}\) = 1.8

4. If two extreme terms of three continued proportional numbers be pqr, \(\frac{pr}{q}\); what is the mean proportional?

Solution:

Let the middle term be x

Therefore, \(\frac{pqr}{x}\) = \(\frac{x}{\frac{pr}{q}}\)

⟹ x\(^{2}\) = pqr × \(\frac{pr}{q}\) = p\(^{2}\)r\(^{2}\)

⟹ x = \(\sqrt{p^{2}r^{2}}\) = pr

Therefore, the mean proportional is pr.

5. Find the third proportional of 36 and 12.

Solution:

If x is the third proportional then 36, 12 and x are continued proportion.

Therefore, \(\frac{36}{12}\) = \(\frac{12}{x}\)

⟹ 36x = 12 × 12

⟹ 36x = 144

⟹ x = \(\frac{144}{36}\)

⟹ x = 4.

6. Find the mean between 7\(\frac{1}{5}\)and 125.

Solution:

The mean proportional of 7\(\frac{1}{5}\)and 125 = +\(\sqrt{\frac{36}{5}\times 125} = +\sqrt{36\times 25}\) = 30

7. If a ≠ b and the duplicate proportion of a + c and b + c is a : b then prove that the mean proportional of a and b is c.

Solution:

The duplicate proportional of (a + c) and (b + c) is (a + c)^2 : (b + c)^2.

Therefore, \(\frac{(a + c)^{2}}{(b + c)^{2}} = \frac{a}{b}\)

⟹ b(a + c)\(^{2}\) = a(b + c)\(^{2}\)

⟹ b (a\(^{2}\) + c\(^{2}\) + 2ac) = a(b\(^{2}\) + c\(^{2}\) + 2bc)

⟹ b (a\(^{2}\) + c\(^{2}\)) = a(b\(^{2}\) + c\(^{2}\))

⟹ ba\(^{2}\) + bc\(^{2}\) = ab\(^{2}\) + ac\(^{2}\)

⟹ ba\(^{2}\) - ab\(^{2}\) = ac\(^{2}\) - bc\(^{2}\)

⟹ ab(a - b) = c\(^{2}\)(a - b)

⟹ ab = c\(^{2}\), [Since, a ≠ b, cancelling a - b]

Therefore, c is mean proportional of a and b.

8. Find the third proportional of 2x^2, 3xy

Solution:

Let the third proportional be k

Therefore, 2x^2, 3xy and k are in continued proportion

Therefore,

\frac{2x^{2}}{3xy} = \frac{3xy}{k}

⟹ 2x\(^{2}\)k = 9x\(^{2}\)y\(^{2}\)

⟹ 2k = 9y\(^{2}\)

⟹ k = \(\frac{9y^{2}}{2}\)

Therefore, the third proportional is \(\frac{9y^{2}}{2}\).

● Ratio and proportion

10th Grade Math

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