a triangle ABC is such that AB= 12cm,<CAB=45degrees and <ABC=60 DEGREES. Calculate the area of triangle ABC
To find the area of triangle ABC, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
In this case, the base of the triangle is AB, which has a length of 12 cm.
To find the height, we can draw a perpendicular line from point B to side AC, creating a right triangle. Let's label this point D.
Since triangle ABC is a 45-45-90 triangle and angle BAC is 45 degrees, angle CAD is also 45 degrees. Thus, triangle CAD is an isosceles right triangle.
In an isosceles right triangle, the length of the perpendicular (or height) from the vertex to the hypotenuse is equal to half the length of the hypotenuse.
Therefore, the height of triangle ABC is half the length of AC, which is equal to half the length of AB.
So, the height (or AD) is (1/2) * 12 cm = 6 cm.
Now we can calculate the area:
Area = (1/2) * base * height
= (1/2) * 12 cm * 6 cm
= 36 cm².
Therefore, the area of triangle ABC is 36 square centimeters.
To calculate the area of triangle ABC, we can use the formula A = (1/2) * base * height.
In this case, the base of the triangle is AB and the height can be found by drawing a perpendicular line from vertex C to side AB.
Since triangle ABC is a right triangle with <CAB = 45°, the height can be calculated using trigonometry.
First, let's find the length of the perpendicular line from vertex C to side AB. This line divides triangle ABC into two right triangles.
Using the sine function, we can find the length of the perpendicular line (height) using the following formula:
sin(<CAB) = Opposite / Hypotenuse
sin(45°) = height / AB
Rearranging the formula, we have:
height = sin(45°) * AB
height = sin(45°) * 12 cm
Calculating sin(45°) ≈ 0.707, we get:
height = 0.707 * 12 cm
height ≈ 8.48 cm
Now, we can calculate the area of triangle ABC using the formula:
A = (1/2) * AB * height
A = (1/2) * 12 cm * 8.48 cm
Calculating this, we find:
A ≈ 50.88 cm^2
Therefore, the area of triangle ABC is approximately 50.88 square centimeters.