given triangle ABC with AB=7cm, BC=8cm and AC=9cm calculate
1. the size of the largest angle
2. the area of the triangle
1.
the largest angle is opposite the largest side, so set your cosine equation up this way:
9^2 = 7^2 + 8^2 - 2(7)(8)cosØ
2.
once you have that angles use the area formula
area = (1/2)(a)(b)sinØ , where Ø is the contained angle.
Find the a triangle whose sides are 7cm,8cm,and 9cm
To solve the given problem, we can use the Law of Cosines to find the largest angle and then use Heron's formula to calculate the area. Let's solve it step by step:
1. Finding the largest angle:
According to the Law of Cosines, the formula is c^2 = a^2 + b^2 - 2ab * cos(C), where c is the side opposite angle C. Applying this formula, we can find the cosine of angle C:
AC^2 = AB^2 + BC^2 - 2(AB)(BC) * cos(C)
9^2 = 7^2 + 8^2 - 2(7)(8) * cos(C)
81 = 49 + 64 - 112 * cos(C)
81 = 113 - 112 * cos(C)
112 * cos(C) = 113 - 81
112 * cos(C) = 32
cos(C) = 32/112
cos(C) = 0.2857
Now, to find the angle, we can use the inverse cosine function (cos^-1) to find the largest angle:
C = cos^-1(0.2857)
C ≈ 75.64°
Therefore, the size of the largest angle, angle C, is approximately 75.64 degrees.
2. Finding the area of the triangle:
We can use Heron's formula to calculate the area of the triangle. Heron's formula states that the area (A) of a triangle with side lengths a, b, and c is given by:
A = sqrt(s(s-a)(s-b)(s-c))
Where s is the semi-perimeter of the triangle, defined as s = (a+b+c)/2.
In this case, let's calculate the area:
s = (AB + BC + AC)/2
= (7 + 8 + 9)/2
= 12
A = sqrt(12(12-7)(12-8)(12-9))
= sqrt(12*5*4*3)
= sqrt(720)
≈ 26.83 square cm
Therefore, the area of the triangle is approximately 26.83 square cm.
To calculate the size of the largest angle in triangle ABC, we can use the Law of Cosines. The formula is as follows:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where:
- c is the length of the side opposite angle C (in this case, AB)
- a and b are the lengths of the other two sides (in this case, BC and AC), respectively
- C is the angle we want to find
1. The size of the largest angle:
Let's plug in the given values into the formula and solve for C:
7^2 = 8^2 + 9^2 - 2 * 8 * 9 * cos(C)
49 = 64 + 81 - 144 * cos(C)
49 = 145 - 144 * cos(C)
144 * cos(C) = 145 - 49
144 * cos(C) = 96
cos(C) = 96/144
cos(C) = 2/3
Now, we need to find the value of C by taking the inverse cosine (cos^-1) of 2/3:
C = cos^-1(2/3)
C ≈ 48.19°
Therefore, the size of the largest angle (angle C) is approximately 48.19°.
2. The area of the triangle:
To find the area of the triangle, we can use Heron's formula, which is suitable for triangles with known side lengths. The formula is as follows:
Area = √(s * (s - a) * (s - b) * (s - c))
Where:
- s is the semi-perimeter of the triangle (s = (a + b + c) / 2)
- a, b, and c are the lengths of the sides of the triangle
By substituting the given values, we can calculate the area:
s = (7 + 8 + 9) / 2 = 24 / 2 = 12 cm
Area = √(12 * (12 - 7) * (12 - 8) * (12 - 9))
Area = √(12 * 5 * 4 * 3)
Area = √(720)
Area ≈ 26.87 cm²
Therefore, the area of the triangle ABC is approximately 26.87 cm².