Can you find the angle X? | (Calculators Not Allowed) |
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- Опубликовано: 13 май 2024
- Learn how to find the angle X. Important Geometry, Trigonometry, and Algebra skills are also explained: area of a triangle formula; SOHCAHTOA. Step-by-step tutorial by PreMath.com
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Algebraic method: if a,b are the other sides of the triangle, ab=1250(2)=2500 and a²+b²=10000, then (a+b)²=10000+2(2500)=15000, (a-b)²=10000-2(2500)=5000, then a+b=50√6, a-b=±50√2; solving equation system: , a=25(√6+√2), b=25(√6-√2) or a=25(√6-√2), b=25(√6+√2), finally tanx=2+√3 or tanx2-√3, and x=75º or x=15º. There are TWO solutions to this problem.
1250=1/2 ×100 sin x × 100 cos x=2500 sin 2x, sin 2x=1/2, 2x=30° or 150°, x=15° or 75°.😊
My method:
Excellent!
From angle B we draw the median BO to the hypotenuse, AO=CO=BO=50;
Notice that ab/2=1250 and a²+b²=100² with tanx=b/a can be transformed to (b/a)/2500=1/a², 1+(b/a)²=100²/a²=100²(b/a)/2500=4(b/a), so tanx²-4tanx+1=0, so tanx=2±√3, and finally tanx=15º or tanx=75º.
One can make a good educated guess that the right triangle is one of those that appear frequently in problems, so we should try those first, especially after being given the clue that calculators are not allowed. Let A = area and h = hypotenuse. So, we try 45°-45°-90°. A = h²/4 = 10000/4 = 2500. Then we try 30°-60°-90°. A = (h²)(√3)/8. Since there is a radical, the area can not be an integer. Next we try 15°-75°-90°, which appears so often in problems that we are familiar with its properties. A = h²/8 = (100)²/8 = 10000/8 = 1250. We have a match! Our answer is x = 15° or 75°.
Well explained
Let AB=a ; BC==b
Being the hypotenuse the base we can find its height =1250*2/100=25
Thank you!
15
I did solve the given Problem this way :
Nice! φ = 30°; ∆ ABC → AC = 100; AB = a; BC = b; CAB = x = ?
Geometric - and fastest - solution: AC is diameter of a circle containing points ABC with center O is at middle of AC with radius r=50. OB is radius too, so OB=r=50. Then AOB is isosceles so ∠ABO=∠OAB=x. Trace height DB from B to AC, then 100DB/2=1250, so DB=25. BDO is a rectangle triangle with hypotenuse OB=r=50 and one cathetus DB=25 then ∠DBO=60º and ∠DOB=30º Depending of position, height DB can be below or above to OB, so
👍
Again I did this the hard way; I solved for 2 equations (A^2+B^2=100^2 and (A*B)/2=1250 using the quadratic equation and denesting the radical.
The answer could be 15 och 75. ( sin(2x) = 0.5 has two solutions: 2x = 30 + n360 or 2x = (180 - 30) + n360 )
Double Angle Identity did the trick!