Find the exact value of each part labeled with a variable in the figure. 21 45 30 r 90 p 45 q p

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Find the exact value of each part labeled with a variable in...

Transcript

00:01 Find the exact value of each label part with the variable in the figure so what we'll do is first of all we'll name the figure that's the easiest way of doing it.

00:09 I'm gonna call this point to be a this is going to be b this is c i have something like d here okay, so abc d form some part and we have some distinct triangle information as well we'll talk about that so in so if you look at in this section the vastest to find the values of p, q and r.

00:37 So with respect to that, we'll look at each of them separately.

00:42 So within the context of this figure, so we can say that in abc, abc is a triangle, abc is a right angled but also a right angled but also isosceles the reason it becomes isosceles triangle is because we look at the angle a and angle c are equal and it is a right angle triangle because it's right angled at b so it's right angled at b which is b being equal to 90 degrees and we have angle a equals to angle c that would make it an isosceles triangle isocles triangle as well so it becomes right angled and isoseless triangle that's the reason it becomes a right angled is hostless triangle the reason i'm illustrating this is because that would make it much easier if you have an isoseless triangle it would then mean the sides are going to be equal which would mean ab in this case would be equal to b c so that's pretty straightforward a b will be equal to you b c ab in this case is 21 and b c is equal to p therefore p is equal to 21 that becomes very easy so that's the first answer uh for p so because they've asked us to work out what p is so that becomes 21 we'll talk about that at the end now then we have to work out the value of r so let's concentrate on this triangle abc so again going back to the triangle a b c we can use the pythagrists theorem and say that the hypotenuse squared which is ac squared will be equal to a b squared plus bc squared now ac squared is equal to r you can see that they've given that ac is equal to r therefore we get r squared is equal to a b is obviously 21 from the information and bc is something that we worked out or it's equal to p we said that p equal to 21 so that it means it's 21 squared as well so r square equals to 21 squared but 21 squared therefore r is square root of 21 squared plus 21 square so r will be equal to 21 root 2 if you did the calculation we will we've actually left it in the integer because they've said that use use integers of fractions so we're not expressing them in decimals so we will leave them in integers and fractions so r is equal to 21 root 2 so that's the second answer that is required as far as this problem is concerned so we'll now concentrate working at q so let's look at the information for q so if you look at this triangle here the q is actually within the triangle acd so we have to concentrate on the triangle acd so as well as triangle acd is concerned c is equal to 90 degrees that's because you can see that this angle here is 90 and therefore in triangle acd actually so say in triangle acd c is 90 degrees therefore we can say that there is a relationship that you can establish between the value of r and value of q in the form of tan because this angle here angle a is also equal to 30 degrees that they've given that so this is within the context of the triangle acd so you can say that considering that triangle acd is a right angle triangle because angle c is at 90 tan 30 degrees would mean it is equal to cd divided by ac cd is nothing but q that's what we are after divided by ac divided is r which we just found out so we can use that information so that would become q is 21 q is basically q so we don't have any information on that divided by r is 21 root 2 so that would mean tan 30 degrees is equal to q divided by 21 square root of 2 but we know tan 30 degrees is actually equal to 1 divided by root 3 this is standard information or you can use a calculator to find it out tan 30 if you calculate the value of tan 30 you get that to be equal to 1 by square root of 3 so what you can do is you can say that because tan 30 is equal to q by 21 root 2 this implies that 1 by square root of 3 that is tan 30 is equal to q divided by 21 square root 2 this implies that 1 by 2 root of 2 so basically q is equal to 21 root 2 divided by root 3 so just cross multiply it so 21 root 2 goes up and then root 3 is just at the denominator q equals to 21 root 3 by 21 root 2 actually i made a mistake apologize so 21 root 2 2 it's 21 root 2.

07:33 It's 21 root 2 divided by root 3.

07:37 Normally it is not a good practice to leave the denominator in the root form.

07:44 We have to rationalize it.

07:46 That's the idea...