Mean Proportional

... and the Altitude and Leg Rules

Mean Proportional

The mean proportional of a and b is the value x here:

a   =   x
x b

"a is to x, as x is to b"

It looks kind of hard to solve, doesn't it?

But when we cross multiply (multiply both sides by b and also by x) we get:

a   =   x
x b
right arrow
ab   =   x
x
right arrow ab  =  x2

And now we can solve for x:

x  =  √(ab)

Example: What is the mean proportional of 2 and 18?

We are being asked "What is the value of x here?"

2   =   x
x 18

"2 is to x, as x is to 18"

We know how to solve it:

x = √(2×18) = √(36) = 6

And this is what we end up with:

2   =   6
6 18

It basically says that 6 is the "multiplication middle" (2 times 3 is 6, 6 times 3 is 18)

mean proportional 2 x3= 6 x3= 18

(It is also the geometric mean of the two numbers.)

One more example so you get the idea:

Example: What is the mean proportional of 5 and 500?

x = √(5×500)

x = √(2500) = 50

So it is like this:

mean proportional 5 x10= 50 x10= 500

 

mean proportional similar triangles inside

Right Angled Triangles

We can use the mean proportional with right angled triangles.

First, an interesting thing:

Those two new triangles are similar to each other, and to the original triangle!

This is because they all have the same three angles.

Try it yourself: cut a right angled triangle from a piece of paper, then cut it through the altitude and see if the pieces are really similar.

We can use this knowledge to solve some things.

In fact we get two rules:

Altitude Rule

The altitude is the mean proportional between the left and right parts of the hyptonuse, like this:

mean proportional left/altitude = altitude/right

Example: Find the height h of the altitude (AD)

mean proportional 4.9 h 10

Use the Altitude Rule:

left   =   altitude
altitude right

Which for us is:

4.9   =   h
h 10

And solve for h:

h2 = 4.9 × 10 = 49
h = √49 = 7

Leg Rule

Each leg of the triangle is the mean proportional between the hypotenuse and the part of the hypotenuse directly below the leg:

mean proportional hyp/leg = leg/part   and   mean proportional hyp/leg = leg/part

Example: What is x (the length of leg AB) ?

mean proportional x 9 7

First find the hypotenuse: BC = BD + DC = 9 + 7 = 16

Now use the Leg Rule:

hypotenuse   =   leg
leg part

Which for us is:

16   =   x
x 9

And solve for x:

x2 = 16 × 9 = 144
x = √144 = 12

Here is a real world example:

mean proportional kite PO is 80, OR is 180

Example: Sam loves kites!

 

Sam wants to make a really big kite:

 

Sam wants to know the length for the strut QS, and also the lengths of each side.

We only need to look at half the kite to do the calculations. Here is the left half rotated 90°

mean proportional triangle p, r, h, 180 and 80

Use the altitude rule to find h:

h2 = 180 × 80 = 14400
h = √14400 = 120 cm

So the full length of the strut QS = 2 × 120 cm = 240 cm

 

The length RP = RO + OP = 180 cm + 80 cm = 260 cm

Now use the Leg Rule to find r (leg QP):

r2 = 260 × 80 = 20800
r = √20800 = 144 cm to nearest cm

 

Use the Leg Rule again to find p (leg QR):

p2 = 260 × 180 = 46800
p = √46800 = 216 cm to nearest cm

 

Tell Sam the strut QS will be 240 cm, and the sides will be 144 cm and 216 cm.

Can't wait for a windy day!