The sies of a triangle are 9cm, 12 cm and 15cm. Find the lenght of the perpendicular drawn to the side which is 15 cm.
Since the sides are in a 3:4:5 ratio, it is a right triangle. The right angle is between the sides with lengths 9 and 12.
The smallest angle is
A = sin^-1(3/5) = 36.87 degrees.
The perpendicular to the hypotenuse is
12 sin A = 12*sin36.87 = 12*0.6 = 7.2 cm
To find the length of the perpendicular drawn to the side which is 15 cm, we can use the formula for the area of a triangle. The area of a triangle can be calculated using the formula:
Area = (1/2) * base * height
In this case, the side which is 15 cm is the base of the triangle. Let's assume the length of the perpendicular as 'h'. So now we have:
Area = (1/2) * 15 cm * h
We need to find the value of 'h', which is the length of the perpendicular. However, we don't have the value of the area. The next step is to calculate the area of the triangle using the given side lengths.
We can use Heron's formula to find the area of the triangle. Heron's formula states that the area of a triangle with sides a, b, and c can be calculated using the semi-perimeter (s) of the triangle. The semi-perimeter is denoted by 's' and can be calculated as:
s = (a + b + c) / 2
Let's calculate the semi-perimeter of the triangle with sides 9 cm, 12 cm, and 15 cm:
s = (9 cm + 12 cm + 15 cm) / 2
s = 36 cm / 2
s = 18 cm
Now, we can use the semi-perimeter (s) and the side lengths to calculate the area of the triangle using Heron's formula. The area (A) can be calculated as:
A = sqrt(s * (s - a) * (s - b) * (s - c))
where sqrt is the square root function.
Let's calculate the area of the triangle:
A = sqrt(18 cm * (18 cm - 9 cm) * (18 cm - 12 cm) * (18 cm - 15 cm))
A = sqrt(18 cm * 9 cm * 6 cm * 3 cm)
A = sqrt(2916 cm^4)
A = 54 cm^2
Now that we have the area of the triangle, we can substitute it back into the formula for the triangle area to find the length of the perpendicular:
54 cm^2 = (1/2) * 15 cm * h
Multiplying both sides by 2:
108 cm^2 = 15 cm * h
Dividing both sides by 15 cm:
h = 108 cm^2 / 15 cm
h = 7.2 cm
Therefore, the length of the perpendicular drawn to the side which is 15 cm is 7.2 cm.