DETERMINE WHETHER THE FOLLOWING MEASUREMENTS PRODUCE A TRIANGLE
About "Determine Whether the Following Measurements Produce a Triangle"
Determine Whether the Following Measurements Produce a Triangle :
Here we are going to see some example problems to determine whether the following measurements produce a triangle.
Determine Whether the Following Measurements Produce a Triangle - Examples
Question 1 :
Determine whether the following measurements produce one triangle, two triangles or no triangle:
∠B = 88°, a = 23, b = 2. Solve if solution exists.
Solution :
Let us apply the given details in sine formula
a/sin A = b/sin B = c/sin C
23/sin A = 2/sin 88° = c/sin C
23/sin A = 2/sin 88°
sin A = (23/2) sin 88°
sin A = 11.5 sin 88°
The maximum value of sin θ will be 1. So, it is not possible.
Hence the given measurements will not produce a triangle.
Question 2 :
If the sides of a triangle ABC are a = 4, b = 6 and c = 8, then show that 4 cosB + 3cosC = 2.
Solution :
Let us use cosine formula to find the values of cos B and cos C.
Cosine formula :
cos A = (b2 + c2 - a2)/2bc --------(1)
cos B = (a2 + c2 - b2)/2ac --------(2)
cos C = (a2 + b2 - c2)/2ab --------(3)
From (2)
cos B = (a2 + c2 - b2)/2ac
cos B = (42 + 82 - 62)/2(4)(8)
cos B = (16 + 64 - 36)/64
cos B = 44/64 = 11/16
From (3)
cos C = (a2 + b2 - c2)/2ab
cos C = (42 + 62 - 82)/2(4)(6)
= (16 + 36 - 64)/48
= -12/48
cos C = - 1/4
Given that :
4 cos B + 3cos C = 2
L.H.S :
= 4(11/16) + 3(-1/4)
= (11/4) - (3/4)
= (11 - 3)/4 = 8/4
= 2 --> R.H.S
Hence it is proved.
Question 3 :
In a triangle ABC, if a = √3 − 1, b = √3 + 1 and C = 60°, find the other side and other two angles
Solution :
cos C = (a2 + b2 - c2)/2ab
a = √3 − 1, b = √3 + 1
a2 = (√3 − 1)2 = 3 + 1 - 2√3 = 4 - 2√3
b = √3 + 1
b2 = (√3 + 1)2 = 3 + 1 + 2√3 = 4 + 2√3
By applying those values in cosine formula
cos 60 = (4 - 2√3 + 4 + 2√3 - c2)/2(√3 − 1)(√3 + 1)
(1/2) = (8 - c2)/2(2)
2 = 8 - c2
c2 = 8 - 2 = 6
c = √6
Sine formula :
a/sin A = b/sin B = c/sin C
(√3 − 1)/sin A = (√3 + 1)/sin B = √6/sin 60°
(√3 + 1)/sin B = √6/(√3/2)
(√3 + 1)/sin B = 2 √6/√3
(√3 + 1)/sin B = 6√2/3
(√3 + 1)/sin B = 2√2
sin B = (√3 + 1)/2√2
sin B = (√3/2) (1/√2) + (1/2)(1/√2)
= sin 60 cos 45 + cos 60 sin 45
= sin (60 + 45)
= sin 105
Hence angle B is 105.
A + B + C = 180
A + 105 + 60 = 180
A + 165 = 180
A = 180-165
A = 15
After having gone through the stuff given above, we hope that the students would have understood, "Determine Whether the Following Measurements Produce a Triangle"
Apart from the stuff given in "Determine Whether the Following Measurements Produce a Triangle", if you need any other stuff in math, please use our google custom search here.
WORD PROBLEMS
HCF and LCM word problems
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
