Name: Proportion
Uploaded: 2019-07-19
Duration: 5 min 17 s
Description: A T-Rex looks strange because it has disproportionately sized arms. But, what are proportions? To understand their significance in mathematics, here are some...

Question

If

y

is the mean proportional between

x

and

z

, then prove that

(x y + y z)

is the mean proportional between

(x^{2} + y^{2})

and

(y^{2} + z^{2})

.

Accepted Answer

A mean proportional is a number $r$ between two numbers $p$ and $q$ , for which $\frac{p}{r} = \frac{r}{q}$ .

Given that $y$ is the mean proportional between $x$ and $z$ , then

$\frac{x}{y} = \frac{y}{z} \Rightarrow y^{2} = x z \Rightarrow y = \sqrt{x z}$ ............. $(i)$

Now, we have to find the mean proportional between $(x^{2} + y^{2})$ and $(y^{2} + z^{2}) .$

Therefore, $\frac{(x^{2} + y^{2})}{Mean proportional} = \frac{Mean proportional}{(y^{2} + z^{2})}$

$\Rightarrow {(Mean proportional)}^{2} = (x^{2} + y^{2}) \times (y^{2} + z^{2})$

$\Rightarrow {(Mean proportional)}^{2} = (x^{2} + x z) \times (x z + z^{2})$ $[$ From $(i)]$

$\Rightarrow {(Mean proportional)}^{2} = x (x + z) \times z (x + z) = x z {(x + z)}^{2}$

$\Rightarrow Mean proportional = \sqrt{x z {(x + z)}^{2}} = (x + z) \sqrt{x z}$

$\Rightarrow Mean proportional = (x + z) y$ $[$ From $(i), \sqrt{x z} = y]$

$\Rightarrow Mean proportional = (x y + y z)$

Hence, proved.

If $y$ is the mean proportional between $x$ and $z$ , then prove that $(x y + y z)$ is the mean proportional between $(x^{2} + y^{2})$ and $(y^{2} + z^{2})$ .

Important Questions on Ratio and Proportion

Learn and Prepare for any exam you want !

If y is the mean proportional between x and z, then prove that (xy+yz) is the mean proportional between x2+y2 and y2+z2.

Important Questions on Ratio and Proportion

Learn and Prepare for any exam you want !

If $y$ is the mean proportional between $x$ and $z$ , then prove that $(x y + y z)$ is the mean proportional between $(x^{2} + y^{2})$ and $(y^{2} + z^{2})$ .