@ricardoguzman5014
Sum of the angles on the base of the triangle sum to 135° because the top angle is 45°. Say that the left angle on the base is θ. Then the right angle is 135°-θ. tan θ=X/3; multiply both sides by 3 3 tan θ=X tan (135°-θ)=X/2; multiply both sides by 2 2 tan (135°-θ)=X X=2 tan (135°-θ) X=3 tan θ, and so: 3 tan θ=2 tan (135°-θ); expand the right side using difference formula for tangent: 3 tan θ=2 (tan 135° - tan θ)/(1 + (tan 135°)(tan θ)) 3 tan θ = 2 (-1-tan θ)/(1 + (-1) tan θ) 3 tan θ = (-2-2 tan θ)/(1- tan θ); multiply both sides by (1- tan θ): 3 tan θ (1- tan θ) = -2 - 2 tan θ 3 tan θ - 3 tan² θ = -2 - 2 tan θ; subtract (3 tan θ - 3 tan² θ) from both sides: 0= -2 - 2 tan θ - 3 tan θ + 3 tan² θ 3 tan² θ - 5 tan θ - 2=0; equation is a quadratic in tan θ, so let tan θ=m 3m² - 5m -2=0. By the quadratic formula, m= (5±√49)/6 Negative value is rejected, so m=(5+√49)/6=(5+7)/6=12/6=2. m=2, and since tan θ=m...tan θ=2 Recalling from above that tan θ=X/3: tan θ=X/3 tan θ=2 Therefore, X/3=2; multiply both sides by 3: X=6
@いまひろ09
x=6 because x=3tanθ1=2tanθ2 ∴ tanθ1=(x/3) tanθ2=(x/2) θ1+θ2=π-π/4=(3/4)π then tan(θ1+θ2)=-1 ∴ (tanθ1+tanθ2)/(1-tanθ1×tanθ2)=-1 i.e. (x/3+x/2)=(x/3)(x/2)-1=(x^2)/6-1 then x^2-5x-6=0 (x-6)(x+1)=0 x=6、-1 x>0 ∴ x=6
@destruidor3003
You make things much more complicated….one only needs to use a tangent to solve this problem
@Raghavverma_16
Here is simple solution We can break the angle 45° as α and 45 - α Now tanα=3/x and tan(45-α)=2/x Now using trigonometry identify of tan(A-B)=(tanA - tanB)/(1+tanA.tanB) => (tan45 - tanα)/(1+tan45tanα)=2/x =>(1-3/x)/(1+3/x)=2/x =>(x-3)/(x+3)=2/x =>x²-5x-6=0 =>x²-6x+x-6=0 =>(x+1)(x-6)=0 Now we get x=1 and x=6 As length cannot be negative so we neglect x=1 So finally x=6
@bryanalexander1839
The only reasoning I could figure with Euclid's theorem is that if the angle were 90° instead of 45° then the height would be √6. In order to maintain a constant 3+2 base: as the angle gets smaller, the height gets taller; and as the angle gets portlier, the height gets shortlier. (Yes, I made up shortlier for the rhyme.) In other words, a height of 1, which is less than √6 requires an angle greater than 90°, so 1 must be rejected since the angle is less than 90°. As others have noted, the obtuse angle that gives a height of 1 is 135° since sin(45°)=sin(135°). My method was to use sum of angles for tangent giving solutions –1 and 6, where extraneous solution –1 is easily identifiable. Since tan(135°)=–tan(45°), were the angle 135° then that would give solutions –6 and 1.
@drahon1
thanks for reminding the Euclids Theorem
@pk2712
sin45 =sin135=1/sqrt2 . x=1 corresponds to angle of 135degees and x=6 corresponds to angle of 45 degrees .
@someone._.5333
What a coincidence... One of my friends sent a math problem involving a rectangle with this exact triangle with the exact angles and length of the side in front of the 45°. Well, the difference being that one demands the area of the triangle, but hey, still need to find the height!
@Dimitar_Stoyanov_359
3:42 Euclid's theorem, as far as I see, is for a right triangle. It's not clear for me how can this be "a hint". As for "why x = 1 can't be a solution", I think it's because 1 < 2 and 1 < 3 (the 2 base parts) and we get angle parts that are both > 45°. Otherwise, we can check the solutions with the tangent sum formula tan(α + β) = (tanα + tanβ) / (1 – tanα·tanβ) α + β = 45° (angle parts at the upper corner) ⇒ tan(45°) = 1 1️⃣ If x = 6 tanα = 3/6 ; tanβ = 2/6 ⇒ (2/6 + 3/6) / (1 – 6/36) = 5/6 / 30/36 = 5·36 / 6·30 = 1 ⇒ Equality isn't broken, x = 6 is a solution. 2️⃣ If x = 1 tanα = 3 ; tanβ = 2 ⇒ (2 + 3) / (1 – 2·3) = – 1 – 1 ≠ 1 ⇒ Equality is broken, x = 1 is not a solution.
@victor1978100
The second root of the equation x^2-5x-6=0 is -1 The side's length can't be negative. So, we left with only one root x=6.
@wolframhuttermann7519
It is easier to use the tangent additional theorem. The other angles add to 135° and their tangent is h/2 and h/3. If you consider tan 135° = - 1, then ever,thing else is easy.
@x.in_hype
Law of cosines 🗿
@Greentangle
180-45=135 for x=1
@ald6980
arctg(1/2)+arctg(1/3)=pi/4 is the common wisdom and can be easily proved. That is why the original problem need no calculations: 3/6=1/2=tg(phi); 2/6=1/3=tg(psi) and the answer is 6.
@noreldenzenky1527
tan 45 = tan(a+b)=(tana+tanb)/(1-tana.tanb) 1=-1 leades contradiction since tan a=2/x ,tanb=3/x where x=1
@somapatra5560
Used inverse tangents
@abasaliabolhassani
Thanks a lot 🎉
@ChristopherBitti
Let’s replace 45 degrees with a variable, say y. It should be obvious that x is a decreasing function of y, meaning if we increase y, we decrease x. For y = 90 degrees, by Euclid’s theorem, we have x = sqrt(6). Thus, we must have x(45 degrees) > x(90 degrees) = sqrt(6) > 1. Thus, x must be 6 when y is 45.
@AJ-fo3hp
22.5+90+67.5=180 67.5+67.5+124=
@RAG981