the rectangle represented the AB side is 4cm the side BC is 6 cm i want to build a triangle taking the AB segment and the third vertice inside the triangle so his area becomes higher than 8 cm*cm.
the area in cm*cm of the region of the rectangle in which can be chosen the third vertice of the triangle is:
D----C
- -
- - 6cm
- -
A----B
4cm
To determine the area in which the third vertex of the triangle can be chosen to make the area greater than 8 cm², we need to consider the possible locations of the third vertex within the rectangle.
Since the rectangle's sides are AB = 4 cm and BC = 6 cm, the rectangle has an area of A = AB * BC = 4 cm * 6 cm = 24 cm².
To find the area in which the third vertex can be chosen to make the triangle's area greater than 8 cm², we subtract the area of the triangle whose area is not greater than 8 cm² from the total rectangle's area.
Let's consider a triangle ABC formed by the sides AB, BC, and AC.
For the triangle's area to be greater than 8 cm², the height from point C to the line AB must be greater than h = 8 cm² / AB = 8 cm² / 4 cm = 2 cm.
The height of the triangle can be visualized as the perpendicular distance between the line AB and the opposite vertex C.
Therefore, the top edge DC can have a maximum length of 2 cm. The length of AD can vary from 0 cm to 4 cm, and the length of BC is fixed at 6 cm.
We can calculate the maximum possible base length AC by considering the right triangle ACD, where AD = x and DC = 2 cm. According to the Pythagorean theorem, AC is given by:
AC² = AD² + DC²
AC² = x² + (2 cm)²
AC² = x² + 4 cm²
Since the length of AD can vary from 0 cm to 4 cm, the maximum possible value of AC is when AD = 4 cm. Substituting this value into the equation above, we get:
AC² = (4 cm)² + 4 cm²
AC² = 16 cm² + 4 cm²
AC² = 20 cm²
To find the area of the region in which the third vertex of the triangle can be chosen, we subtract the area of the triangle ABC from the total area of the rectangle.
The area of the triangle ABC is given by:
Area = (1/2) * base * height
Area = (1/2) * AC * DC
Area = (1/2) * √20 cm * 2 cm
Area = √20 cm²
Therefore, the area in cm² of the region in which the third vertex of the triangle can be chosen is:
Area = A - √20 cm²
Area = 24 cm² - √20 cm²
Area = 24 cm² - 4.47 cm²
Area ≈ 19.53 cm²
So, the area in cm² of the region in which the third vertex of the triangle can be chosen to make the triangle's area greater than 8 cm² is approximately 19.53 cm².