Determine how many triangles with the following measurements can be formed: �ÚA = 53�‹, a = 7, b = 10.
HELPP!!
Draw a triangle ABC with AB the base, and C the vertex.
You can see that C can be acute or obtuse, allowing two triangles to be constructed with base = 10 angle A=53°, and a=7.
That can also be see from the law of cosines:
c^2 = a^2 + b^2 - 2ab*cosC
cosC can be positive or negative, depending on whether C is acute or obtuse.
To determine how many triangles can be formed with the given measurements, we can use the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we have the following measurements:
- Side A (opposite angle A): A = 53°
- Side a: a = 7
- Side b: b = 10
To apply the triangle inequality theorem, we need to consider the three possible combinations of sides:
1. Side A and side a:
The sum of side A and side a cannot be less than side b or equal to it.
A + a > b
53° + 7 > 10
60° > 10 (which is true)
2. Side A and side b:
The sum of side A and side b cannot be less than side a or equal to it.
A + b > a
53° + 10 > 7
63° > 7 (which is true)
3. Side a and side b:
The sum of side a and side b cannot be less than side A or equal to it.
a + b > A
7 + 10 > 53°
17 > 53° (which is false)
As a result, we can form a triangle with the given measurements. The answer is: 1 triangle can be formed.