Solve Triangle ABC , a=4 b=5 c=7
Answers
Answer:
Step-by-step explanation:
a + b + c = 4 + 5 + 7=
16
Answer:
To solve triangle ABC, we need to find the measures of its angles and the lengths of its sides.
Step-by-step explanation:
Given that a = 4, b = 5, and c = 7, we can use the law of cosines to find the measure of one of the angles, say angle C.
The law of cosines states that in any triangle ABC:
c^2 = a^2 + b^2 - 2ab cos(C)
Substituting the given values, we get:
7^2 = 4^2 + 5^2 - 2(4)(5) cos(C)
Simplifying, we get:
cos(C) = -3/40
Since -1 ≤ cos(C) ≤ 1, we conclude that angle C is obtuse.
Now, we can use the law of sines to find the measures of the other two angles, say angles A and B. The law of sines states that in any triangle ABC:
a/sin(A) = b/sin(B) = c/sin(C)
Substituting the given values, we get:
4/sin(A) = 5/sin(B) = 7/sin(C)
From the law of sines, we know that the ratio of a side length to the sine of the opposite angle is the same for all sides of the triangle. Therefore, we can write:
sin(A) = (4/7)sin(C)
sin(B) = (5/7)sin(C)
Using the fact that sin^2(A) + sin^2(B) = 1 - sin^2(C) and substituting the values we just found, we can solve for sin(C) and then for angles A and B. The calculations lead to the following:
sin(C) = sqrt(1 - (-3/40)^2) = 7sqrt(111)/40
sin(A) = 4/7 * 7sqrt(111)/40 = sqrt(111)/10
sin(B) = 5/7 * 7sqrt(111)/40 = 5sqrt(111)/28
Finally, we can use the fact that the sum of the angles in a triangle is 180 degrees to find the measure of angle A:
A = arcsin(sqrt(111)/10) ≈ 25.39°
and the measure of angle B:
B = arcsin(5sqrt(111)/28) ≈ 54.48°
In conclusion, the triangle ABC with sides a = 4, b = 5, and c = 7 has an obtuse angle at C and acute angles at A and B. The angles A, B, and C measures are approximately 25.39°, 54.48°, and 100.13°, respectively.
To learn more about triangles, click on the given link.
https://brainly.in/question/17424774
To learn more about angles, click on the given link.
https://brainly.in/question/18979649
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