Find the Value of X & Y in this Right Triangle | Fast & Easy Explanation

[AnonimityAssured]

That seemed a lot of work considering that it is easy to show that the yellow triangle is isosceles and that CD therefore has a length of 2 units.

[georgecaplin]

The equation he worked on from 4:45. Geez he made that hard work. Divide both sides by sq. rt. of 3, distribute the fraction. He hung on to that square root to the bitter end.

[harikatragadda]

Its better to use the Exterior Angle Theorem and solve it simply in two steps.
Step1: ∠DCA=30°
∴ CD= DB= 2
Step2: AD= CD*Cos60=1
AC= AC*Sin60=√3

[ebich]

∠ACB=60, ∠ACD=30, hence, ∠DCB=30, hence, △CDB is an isosceles triangle, hence, DB=CD, CD=2, hence, AD=1, AB=√3

[interestingmath]

2 nd method is
In triangle BCD, angle 60 degree is equal to two interior angles i.e. 30 + angle C therefore 60= 30+ angleC
And We get angle C = 30 therefore BD=CD=2(if two angle are equal then there side is also equal)
Now in triangle ACD, CD=2 therefore
Sin60=y/2, therefore we get y= √3
And cos60=x/2, therefore x=1

[py]

Took less than 15 seconds of mental calculations to find the answers. After immediately recognising CD is equal to BD which is 2, it’s then just a matter of applying cos60’=1/2 and sin60’=sqrt(3)/2.

[kennethbautista]

Or you can use line CD
Since Triangle BCD has 2 equal angles, therefore the two sides must be equal too.
Line CD = Line DB = 2
Use triangle ACD
y = 2sin(60) = sqrt 3
x = 2 cos(60) = 1

[murdock]

Nice, many thanks, Sir!
φ = 30° → sin⁡(φ) = x/2 = 1/2 → x = 1 → AB = 3 → y = √3

[wwoods]

In Step 3, at 5:00, you could save a bit by multiplying both sides by √3.
√3 x = (√3/3)(x+2) --> 3x = (3/3)(x+2) = x+2 --> 2x = 2.

[mattiaserturini]

Hi, to solve this problem I imagined CD = x and CB = y, so the answers will be x sin (60 °) and x cos (60 °).
After doing the system
{y sin (30 °) = x sin (60 °)
{2 + x cos (60 °) = y cos (30 °)
I find x = 2 and y = 2√3
After this I do 2sin (60 °) = √3 and 2 cos (60 °) = 1

[mustafizrahman]

X=1 unit and Y= root 3.Thanks for giving this problem, Sir.

[robertlynch]

I found the correct answer without using sin() or cos() at all. Simple Pythagoras arithmetic, actually:

№ 1.1: base = (𝒙 + 2)
№ 1.2: height = (𝒚);

The hypotenuses of the △s

№ 2.1: 𝒉² = (𝒙 + 2)² + 𝒚² (outer)
№ 2.2: 𝒋² = 𝒙² + 𝒚²

Recognizing that there are 30°–60°–90° triangles involved, which haver a ratio of 1:2 between shortest side and hypotenuse, it also can be said:

№ 3.1: 𝒉 = 2𝒚
№ 3.2: 𝒋 = 2𝒙

Substituting those into № 2.1 and 2.2:

№ 2.1a: (2𝒚)² = (𝒙 + 2)² + 𝒚² … expanding
№ 2.1b: 4𝒚² = 𝒙² + 4𝒙 + 4 + 𝒚² … moving 𝒚
№ 2.1c: 3𝒚² = 𝒙² + 4𝒙 + 4

№ 2.2a: (2𝒙)² = 𝒙² + 𝒚² … expanding
№ 2.2b: 4𝒙² = 𝒙² + 𝒚² … moving 𝒙
№ 2.2c: 3𝒙² = 𝒚²

With № 2.2c, we have a substitution into № 2.1c that gives us 𝒙 only:

№ 4.1: 3(3𝒙²) = 𝒙² + 4𝒙 + 4 … expanding
№ 4.2: 9𝒙² = 𝒙² + 4𝒙 + 4 … moving 𝒙²
№ 4.3: 8𝒙² = 4𝒙 + 4 … moving rest of terms
№ 4.4: 8𝒙² - 4𝒙 - 4 = 0 … giving quadratic form

№ 4.5: 𝒙 = (4 ± √(16 + 4×8×4))/(2×8)
№ 4.6: 𝒙 = (4 ± √144)/(16)
№ 4.7: 𝒙 = (4 ± 12)/(16)
№ 4.8: 𝒙 = [1, –½]

Obviously 𝒙 shouldn't be negative, so (𝒙 = 1) is the first solution. Using № 2.2c, we get

№ 2.2d: 3×1² = 𝒚²
№ 2.2d: √3 = 𝒚

And there are the goals, complete.

⋅-⋅-⋅ Just saying, ⋅-⋅-⋅
⋅-=≡ GoatGuy ✓ ≡=-⋅

[miltonluoma]

Here is another approach:

Angle ACB is 60 degrees because angle ABC is 30 and CAB is 90
Angle ACD is 30 degrees because angle ADC is 60
Therefore, angle DCB is 30 degrees, therefore the yellow triangle is isosceles and CD = 2
Y^2 + X^2 = 4
X = half of CD = 1
Y^2 = 4 - 1 = 3
Therefore Y = square root of 3

[mathsandsciencechannel]

Very great video sir. Helping student gain confident in math. Thank you sir. I admire you a lot

[najigharib]

Thank you for your effort
I may find the values in sach a way:
As ABC is semi equilateral triangle then angle(ABC)=30° and angle(ACB)=60°
as angle(ADC)=60° then ADC is semi equilateral triangle and angle(ACD)=30° and then angle(DCB)=30°
Then angle(DCB)=angle(DBC)=30° and the triangle DCB is isosceles and then DC=DB=2
in the semi equilateral triangle ADC we have AD=(CD/2)=2/2=1 and then x=1
Applying Pythagoras theorem in the triangle ADC we get y^2+x^2=2^2=4 then y^2= 4-x^2=4-1=3 then y= sqrt(3)

[seroujghazarian]

angle DCA=30°
angle ACB=60°
<=> angle DCB=30°=angle DBC
thus the triangle BCD is isoceles and CD=BD=2
<=> x^2+y^2=4

x^2+y^2+4x+4=4+4+8/2
4+4x+4=4+4+4
4x=4
x=1
1+y^2=4
y^2=3
y=sqrt(3)

[kennethstevenson]

You can fill in all the missing angles. The triangle on the right is an isosceles triangle with equal sides of 2 opposite equal 30 degree angles. Triangle ADC is a 30, 60, 90 triangle with sides equal to 1, 2, root 3 since its longest side is 2. Therefore X=1 and Y= root 3.

[rajeshreddy]

Hi sir I started doing Math league they gave me questions like these so I needed to study them because I am very bad at them now that I found your channel I have started watching your videos you explain so well please keep on uploading videos like these thank you sir

[Wildcard]

trying to solve it on my own before watching

1. The angles at C are 30° for red and 60° for total - so the yellow part is also 30°
2. The yellow triangle has to be isoscaled, so the distance CD is 2.
3. sin 30° is 1/2, so x/2 is 1/2, so x=1
4. x²+y²=2² ; that means y²=2²-x² =4-1 = 3, so y=sqrt(3).

(editing reason: vocabulary trouble)

[devondevon]

The yellow triangle is an isosceles triangle since CDB=120 degree hence DBC=30 degree, hence the other side is 2 since the pink triangle is 30, 60, 90 then y is sqrt 3 and x is 1 as 30, 60 90 degree triangle sides are x, 2x(the hypotenuse) and sqrt 3x.
Answer y= sqrt 3 and x=1