AB = 8 cm, AC = 6 cm, AD = 7 cm, CD = 2.82 and CAB=50° (a) the length BC (b) The size of angle ABC; (c) The size of angle CAD (d) The area of triangle ACD
To find the answers to these questions, we can use various properties and formulas related to triangles. Let's go through the step-by-step process for each part of the question:
(a) The length of BC:
To find the length of BC, we'll use the law of cosines, which states that in a triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice their product multiplied by the cosine of the included angle.
In this case, we can use the law of cosines to find BC:
BC^2 = AB^2 + AC^2 - 2 * AB * AC * cos(CAB)
Substituting the given values:
BC^2 = 8^2 + 6^2 - 2 * 8 * 6 * cos(50°)
BC^2 = 64 + 36 - 96 * cos(50°)
BC^2 = 100 - 96 * cos(50°)
Now, we can calculate BC:
BC = sqrt(100 - 96 * cos(50°))
(b) The size of angle ABC:
To find the size of angle ABC, we can use the law of sines, which states that the ratio of the length of a side to the sine of the angle opposite that side is constant for all sides and angles in a triangle.
In this case, we can use the law of sines to find angle ABC:
sin(ABC) / BC = sin(CAB) / AB
Substituting the given values:
sin(ABC) / BC = sin(50°) / 8
Now, we can solve for sin(ABC) and then find angle ABC:
sin(ABC) = (sin(50°) / 8) * BC
angle ABC = arcsin((sin(50°) / 8) * BC)
(c) The size of angle CAD:
To find the size of angle CAD, we can use the law of cosines again:
cos(CAD) = (AD^2 + CD^2 - AC^2) / (2 * AD * CD)
Substituting the given values:
cos(CAD) = (7^2 + 2.82^2 - 6^2) / (2 * 7 * 2.82)
Now, we can solve for CAD:
CAD = arccos((7^2 + 2.82^2 - 6^2) / (2 * 7 * 2.82))
(d) The area of triangle ACD:
To find the area of triangle ACD, we'll use Heron's formula, which states that the area of a triangle can be calculated using only the lengths of its sides.
In this case, we have the lengths AC, CD, and AD:
s = (AC + CD + AD) / 2
area = sqrt(s * (s - AC) * (s - CD) * (s - AD))
Substituting the given values and using the calculated values from previous steps, we can find the area of triangle ACD.
I hope this step-by-step explanation helps you find the answers to your questions! Let me know if you need any further assistance.