Geometry question

[pliant knot]

Could someone explain me how they'd solve the a)?

Thanks in advance

[pliant knot]

Using Thales theorem, we know that D is a right angle, thus we can use the theorem of Pythagoras to find out that the line DB is sqrt(13).
To find the sin(A) I need to do opposite/hypotenuse, the opposite is 4 (CB) but the hypotenuse is sqrt(13) + AD, but I dont know what AD is

The solution of this exercise is given: sqrt(13)/4, but I dont understand how to get to it

[willow storm]

notice that triangles ACD and BCD have two angles in common

[pliant knot]

I see that they have D in common, what is the 2nd?

[willow storm]

sorry, typo: I meant ABC and BCD (though funnily enough I wasn't actually wrong)

[pliant knot]

They only have B in common, don't they?

They have the vertex C in common as well, but not the angle

[willow storm]

I mean that they both have right angles

[pliant knot]

I see, but how does this help me?

[willow storm]

that means that the third angle is the same as well

so they're similar triangles

[pliant knot]

Ah and the ratios of sin are the same then, correct?

Would I need to do a cos(C) then, since the C angle is the same as A for ABC?

[willow storm]

the most obvious thing to do for me is find the length of AB

it's possible I'm missing a trick, given the order of the questions

[pliant knot]

How would I find the length of AB by knowing that ABC and CDB are similar?

[willow storm]

the ratios of the sides are the same

so for example AB : BC = BC : BD

[pliant knot]

I understand that since ABC and DCB are similar, they are differently scaled versions of each other, right?

[pliant knot]

I am not sure if I understand this notation, do you mean division by : ?

[willow storm]

it's ratio notation, but yes, you can think of it as division in this case

if I say a : b : c = d : e : f then a/b = d/e, a/c = d/f, b/c = e/f

[pliant knot]

the angle of A is the same as C for CBD, right?

[pliant knot]

I see, thanks

[willow storm]

yep, because both add up to 90 with the B angle

[pliant knot]

I still struggle to find the AB length by knowing this though

[pliant knot]

Wouldnt AB : BC = DB : BC, since AB = c*DB?

[willow storm]

that's saying AB = DB

notice that BC is the hypotenuse of the smaller triangle but a leg of the larger one

[pliant knot]

I see, so the hypotenuse AB is = CB in the ratio

[pliant knot]

So if AB/CB = CB/BD, then AB*BD=(CB)² and AB=CB²/BD, thus when filling in the values AB=16/sqrt(13), right?

[willow storm]

sounds about right

[pliant knot]

And then I have 4²+x²=(16/sqrt(13))² and solving for x leads to x² = 256/13 - 16 and x = sqrt(48/13) = AC

Then I could calculate sin(A) as opposite/hypotenuse which would mean 4/(16/sqrt(13)) which is the correct solution.

Thanks for your help!

[willow storm]

np!

[jovial canopy]

I dunno what you guys just said, but this is a better approach (for me) just wanted to help!

[willow storm]

oh yeah, you can just take the sin of angle BCD lol

[willow storm]

(we also noted that all the triangles were similar, but found the side lengths of the larger triangle directly)

[turbid oriole]

.close