Hey thanks mate! very well explained thank you.

I did have a few other questions: have attached them below

**5c.** Whenever we see similar triangles, we know that the ratio of one side to the other of one of the triangles is going to be exactly the same for the other triangle. Using this rule, we know that AB:BC = DE:EF.

Once we know this rule, it becomes clear what we have to do. We only have x as a pronumeral and 3 as a constant, so there will be no need for simultaneous equations here. If AB:BC = DE:EF, just plug in these numbers as fractions. In other words, we set up an equation that shows the ratios are equivalent, using fractions.

So, x/(x+2) i.e AB/BC is equal to 3/x i.e DE/EF.

Now that we know x/(x+2) = 3/x, solve manually or plug it into the CAS (seems like most of these are CAS questions) to get x = (√33+3)/2

**7.** This questions may look complex from that first equation but it is actually pretty simple. We just need to set up an equation that links everything together. We know that AB = 8, so then we can assume that BP = the entire line (AP) - AB (8 ), or BP = AP - 8.

Now, it gives us AB * AP = BP

^{2}, which may look harder to solve because we only know AB, which is 8. However, if we can eliminate one of those pronumerals completely, we will only have one variable to solve making it much simpler.

This becomes clear when considering our linking expression, BP=AP-8. If we can express BP in terms of AP, this completely eliminates the need for BP and slims down the given expression AB * AP = BP

^{2} to only one variable. So, substituting BP with AP-8, we get AB * AP = (AP-8)

^{2}. We know that AB=8, so 8 * AP = (AP-8 )

^{2}. Switch out AP with something simple like

**a** for the CAS and we get 8*a=(a-8)

^{2}. Solve it for a and we get AP = 4(√5+3).

Now for BP, we already have an expression for that so its much simpler to solve. Just sub AP (which is 4(√5+3)) into BP = AP - 8 and we get BP = 4√5+4.

**9.** If I'm interpreting this question right, it is asking us to form two equations which will then be solved simultaneously.

Let the first number be

**a**, and the second number be

**b**. We don't know

** a** or

**b**, the only thing we know is that b is going to be 5 more or less than a, so we can assume that b = a+5.

Then, we know the sum of their squares is 100. In other words, a

^{2} + b

^{2} = 100.

So we have two equations:

a+5=b

a

^{2} + b

^{2} = 100.

Solve simultaneously using CAS and we get a=4.11438 and b=9.11438.

Hope this helps!