Find the height h in the right triangle | Important Geometry and Trigonometry skills explained

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Find the height h in the right triangle | Important Geometry and Trigonometry skills explained

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[bigm]

Again, an excellent, stepwise explanation. Thanks Professor!

[philipkudrna]

I derived the following formula for which yields h=127, 63.

[geoffreyparfitt]

Perhaps simpler to use tangents. Giving h = 90tan(66)tan(41)/(tan(66) - tan(41)) = 127.63 units.

[ZombieKillerThe]

You can also use nothing but tan trig function

Remember tan(x)=opp/adj => adj*tan(x)=opp

h=(90+x)*tan(41)=x*tan(66)
90tan(41)+xtan(41)=xtan(66)
90tan(41)=xtan(66)-xtan(41)
90tan(41)=x(tan(66)-tan(41))
x=90tan(41)/(tan(66)-tan(41))

h≈127.634278999

PS I try to avoid rounding until the end for more accurate answer. Plus I like working out exact value of something like h is exactly

Also Line BD is units

[philippeganty]

Why not consider the tangents of angles D and A, with DB = x? In this case, tang 66° = h/x and tang 41° = h/(x+90).
So (x+90)/x = tan 66°/tan 41°.
By expanding, we find x = 90/((tan 66°/tan 41°)-1).
As h = x(tang 66°), the answer is:

[jimlocke]

At about 4:00, to simplify the solution method, make use of sin(x) is equal to opposite side/hypotenuse. So, sin(41°) = h/(194.47) Therefore, h ~= 194.47 ( sin 41°) ~= 127.57

[arkadipray]

We can use h = d/(cot x - cot y) where d is distance supplied 90 and h is height

Keeping in mind x <y
Or,
h = d/(tan x - tan y) where x>y

[loneranger]

Dear Professor your explanations are unparalleled and par excellent so that most of the people can fully understand.

[AnonimityAssured]

The actual digits after the decimal point are 634..., which suggests that you are rounding too early in the calculation. Still, it's the method that counts. It can be used very effectively to measure the height of a distant structure that you can't reach.

[royalmilk]

Thanks to you i improved my marks from 36% to 88%, thank you so much PREMATH!!!

[murdock]

Awesome question, excellent way to solve the problem, many thanks!
A slightly different approach:

AD = 90
BD = x
BC = y →
tan⁡(φ) = y/(x + 90)
tan⁡(γ) = y/x →
x/(x + 90) = tan⁡(φ)/tan⁡(γ) ≔ m →
x = 90m/(1 - m) ≈ 56, 86 → 90 + x = 146, 82 → tan⁡(φ) = y/146, 82 →
y = (146, 82)tan⁡(φ) ≈ 127, 63 🙂

[j.r.]

I agree with the leading comment below. Always use the tangent function when you're given either the opposite or adjacent side length (or as here, at least part of one of those) and you need to find the other one.

[benjamindomingo]

I solved it this way: let DB is =x Therefore the first equation is h/x =tan 66 or h=x tan 66 and the 2nd equation is h/90 + x = tan of 41. Now we have 2 variables ( h, x)and 2 equations. By substituting h= x tan 66 to the 2nd equation and proceed the answer is 127.56 which is the same.

[joansoldevilacaba]

There is no need to find lthe length AD. Instead, you can find the length dc. It is equal to 139.71, and then use the angle of 66 degrees applying the law of sines. The result is mainly the same. I applied this to find the solution👍👍👍

[MrPaulc]

Excellent video. I didn't calculate AC. I went 90/sin(25) = CD/sin(41) which gives CD as 139.713... then sin(66) = h/139.713... from which h can be calculated. I ended up with 127.634... for h, but the discrepancy is because I didn't round until the very end.

[suchitroybr]

Can easily find if you use tangents alone
if y is the base of smaller right triangle,
From the larger right triangle,
tan41 = h/(90 + y) OR h = tan 41(90 + y)
h = 0.8693 (90 + y)
h = 78.2358 + 0.8693y ....(1)
From the smaller right triangle,
tan66 = h/y OR h = tan66 . y
h = 2.2460y....(2)
From (1) and (2),
78.2358 + 0.8693y = 2.2460y
2.2460y - 0.8693y = 78.2358
1.3767y = 78.2358
y = 56.8285
But from (2), h = 2.2460y
h = 2.2460 x 56.8285
h = 127.6368

[alster]

I used the tan method aka Angle of Elevation/Depression method and got the same answer

[guyhoghton]

From △BCD cot 66° = DB/h.
From △ABC cot 41° = (90 + DB)/h = 90/h + cot 66°.
∴ h = 90 / (cot 41° - cot 66°).

[kostaskatsas]

koka formula h=k(cotb-cota) , so h=90(c0t61-cot41) aproximately 127.566 units

[mathsplus]

Beautiful problem and explanation. I use the common height method but this method is just a good. Perfect presentation Sir