A triangular piece of wood having a dimension 130 cm, 180 cm, and 190 cm is to be divided by a line bisecting the longest side drawn from its opposite vertex. Find the area of the part adjacent to 180 cm.  Group of answer choices5613 sq. cm.5162 sq. cm.5126 sq. cm.5216 sq. cm.

To find the area of the part adjacent to the side measuring 180 cm, we need to bisect the longest side drawn from its opposite vertex.

Step 1: Identify the longest side of the triangular piece of wood. In this case, the longest side is 190 cm.

Step 2: Draw a line from the opposite vertex of the longest side to bisect it. This line will divide the triangle into two parts.

Step 3: Calculate the area of the part adjacent to the side measuring 180 cm. To do this, we need to find the area of the entire triangle and then subtract the area of the other part.

Step 4: Use Heron's formula to find the area of the entire triangle. Heron's formula states that the area of a triangle with sides a, b, and c is given by the formula:

Area = √(s(s-a)(s-b)(s-c))

where s is the semi-perimeter of the triangle, given by the formula:

s = (a + b + c) / 2

In this case, the sides of the triangle are a = 130 cm, b = 180 cm, and c = 190 cm. Plugging these values into the formula, we get:

s = (130 + 180 + 190) / 2 = 250 cm

Area = √(250(250-130)(250-180)(250-190)) = √(250(120)(70)(60)) = √(126000000) = 11228.5 sq. cm

Step 5: Now, we need to find the area of the other part of the triangle. Since the line bisects the longest side, the other part will be a right-angled triangle.

Step 6: Use the formula for the area of a right-angled triangle, which is given by the formula:

Area = (1/2) * base * height

In this case, the base of the right-angled triangle is 180 cm and the height can be found using the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Using the Pythagorean theorem, we can find the height of the right-angled triangle:

height^2 = 190^2 - 180^2 = 36100 - 32400 = 3700

height = √3700 = 60.83 cm

Area = (1/2) * 180 * 60.83 = 5497.4 sq. cm

Step 7: Finally, subtract the area of the other part from the area of the entire triangle to find the area of the part adjacent to the side measuring 180 cm:

Area of part adjacent to 180 cm = 11228.5 - 5497.4 = 5731.1 sq. cm

Therefore, the area of the part adjacent to 180 cm is approximately 5731.1 sq. cm.

Based on the given answer choices, the closest option is 5613 sq. cm.