what is x in a 30-60-90 triangle?

Question

if the side opposite the 30 degree angle is 3 and the side opposite 60 degree angle is x+4 then wat is x?

Answer 1

GIVEN:
30° - 60° - 90° ▲
The side opposite the 30° angle = 3
The side opposite the 60° angle = (x + 4)

FIND:
The value of "x".
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REMEMBER:
The smallest ratio of the sides of a 30° - 60° - 90° ▲ is:
1 : √(3) : 2
The side opposite the 30° angle = 1
The side opposite the 60° angle = √(3)
The side opposite the 90° angle = 2
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To find the side opposite the 90° angle,
MULTIPLY the side opposite the 30° angle by 3 to get its side-ratio of 3.
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MULTIPLYING the side opposite the 30° angle of 1 by 3 we get:
1(3) = 3 units

Therefore, MULTIPLY the side opposite the 90° angle by 3 to get its unit length.
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MULTIPLYING the side opposite the 90° angle of 2 by 3 we get:
2(3) = 6 units

Now, we can use the PYTHAGOREAN THEOREM to SOLVE for "x".
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REMEMBER:
THE PYTHAGOREAN THEOREM is as follows. . .
a² + b² = c²
(the side opposite the 30°)² + (the side opposite the 60°)² = (the side opposite the 90°)²
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The side opposite the 30° angle = a = 3
The side opposite the 60° angle = b = (x+4)
The side opposite the 90° angle = c =6

a² + b² = c²
(3)² + (x+4)² = (6)²
9 + [(x+4)(x+4)] = 36
9 + x² + 4x + 4x + 16 = 36
x² + 8x + 25 = 36
x² + 8x + 25 - 25 = 36 - 25
x² + 8x + 0 = 11
x² + 8x = 11
x² + 8x - 11 = 11 - 11
x² + 8x - 11 = 0

This Trinomial is PRIME, so we must use the QUADTRATIC FORMULA to find the value(s) for "x".
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REMEMBER:
STANDARD FORM for a QUADRATIC EQUATION is as follows. . .

ax² + bx + c = 0

The QUADRATIC FORMULA is as follows. . .

x = [-b ±√(b² - 4ac)] / (2a)

Where "a", "b", and "c" are leading coefficients and a≠0.
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x² + 8x - 11 = 0
1x² + 8x - 11 = 0
ax² + bx + c = 0

Therefore,
a = 1
b = 8
c = -11

SUBSTITUTE the above values using (PARENTHESIS) into the QUADRATIC FORMULA, to find the "x" value(s) for the above question.
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LET:
a = 1
b = 8
c = -11

SUBSTITUTION:
x = [-b ±√(b² - 4ac)] / (2a)
x = { -(8) ±√[(8)² - 4(1)(-11)] } / [(2)(1)]

SIMPLIFY by doing any EXPONENTS, first from L → R.
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x = { -(8) ±√[(8)² - 4(1)(-11)] } / [(2)(1)]
x = { -(8) ±√[(64) - 4(1)(-11)] } / (2)

Where there are (PARENTHESIS). . .
There is Multiplication. SIMPLIFY by MULTIPLYING from L → R.
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x = { -(8) ±√[(64) - 4(1)(-11)] } / (2)
x = { -1(8) ±√[1(64) - 4(1)(-11)] } / 1(2)
x = { -8 ±√[64 - 4(-11)] } / 2

SIMPLIFY by MULTIPLYING from L → R, again.
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x = { -8 ±√[64 - 4(-11)] } / 2
x = [ -8 ±√(64 + 44) ] / 2

SIMPLIFY by ADDING "like" terms within the (PARENTHESIS).
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x = [ -8 ±√(64 + 44) ] / 2 .....Ah HA! I found my mistake.
...............................................I accidentally got a sum of 110
..............................................when ADDING 64 + 44.
..............................................64 + 44 =108
x = [ -8 ±√(108) ] / 2 ..................YES. That's MUCH better. ;o)

SIMPLIFY the radical.
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x = [ -8 ±√(108) ] / 2
x = { -8 ±√[(36)(3)] } / 2
x = { -8 ± [√(36)][√(3)] } / 2
x = { -8 ± 6√(3) } / 2

Factor out the GCF from BOTH the numerator and the denominator.
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x = [ -8 ± 6√(3) ] / 2
GCF =2

x = [ -8 ± 6√(3) ] / 2
x = 2[ -4 ± 3√(3) ] / 2(1)

REDUCE.
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x = 2[ -4 ± 3√(3) ] / 2(1)
x = [ -4 ± 3√(3) ] / 1
x = -4 + 3√(3)

Lastly, find both approximate values for "x".
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x = -4 + 3√(3)
x ≈ -4 + 5.196152423
x ≈ 1.196152423

x = -4 - 3√(3)
x ≈ -4 - 5.19652423
x ≈ -9.196152423

However, there is a RESTRICTION on "x".
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REMEMBER:
Any side length of a polygon, MUST be > 0.
Therefore, the following statement MUST be TRUE: x + 4 > 0
for ALL values of "x".
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RESTRICTION on "x":
x + 4 > 0
x + 4 - 4 > 0 - 4
x + 0 > -4
x > -4

Therefore, "x" MUST be Greater Than (-4).
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Since x > -4

x ≈ 1.196152423
1.196152423 > -4 is a TRUE statement.
Therefore, x ≈ 1.196152423 is a solution for "x".

However,
x ≈ -9.196152423
-9.196152423 > -4 is a FALSE statement.
Therefore, x ≈ -9.196152423 in NOT a solution for "x".
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FINAL ANSWER:

x = -4 + 3√(3)

x ≈ 1.196152423

GREAT question! :o)
Congrats. on making it to the end with me! ♥
I even made a few mistakes along the way :o}

Answer 2

if the side opposite of the 30 degree angle is 3, then the side opposite of the 60 degree angle is 3radical3, and the side opposite of the 90 degree angle is 6.
you use this format
opposite 30 = x
opposite 60 = xradical3
opposite 90 = 2x
this is for simple triange

you can use tangent for this. tan60=(x+4)/3 remember s o h c a h t o a.

Answer 3

Sorry to see the math professor get lost. Here's a purely geometric approach by an astrophysicist (try saying that with peanut butter in your mouth). NO TRIG. WHATSOEVER. I'll do EVERY step!:

Reflect the triangle wrt to the longest perpendicular side. You now have a large equiangular and therefore equilateral triangle --- all angles 60 deg., so all sides are equal. Therefore that common length for the sides of this larger triangle is 2 x 3 = 6.

Back to the original triangle. You now know its hypoteneuse is
6. Therefore, by Pythagoras's Theorem, the length of the longest perp. side is sqrt (6^2 - 3^2) = sqrt 27 = 3 sqrt 3.

So the given length, x + 4 = 3 sqrt 3.

Hence x = (3 sqrt 3) - 4.

(Bracket solely to avoid the slightest ambiguity.)

Isn't it far more satisfying and aesthetic to solve it for oneself from geometric first principles than relying on trig.? I think so.

Live long and prosper.

Answer 4

3/(sen 30) = (x+4)/(sen 60)
3/(1/2) = (x+4)/[(3^¹/2)/2]
3 = (x+4)/(3^¹/2)
3(3^¹/2) = x+4

x = 3(3^¹/2) - 4

x = 3*sqrt(3) - 4

₢

Answer 5

x + 4 = 3*sqrt3
x = 3*sqrt3 - 4

Answer 6

x = 3(tan60) - 4 = 1.196152...