A triangle has vertices (3, 0) and (6, 4). One of its angles is bisected by the x axis. Find the perimeter of the triangle.

since (3,0) is on the x-axis, the angle there must be the one bisected by the axis. So, the 3rd vertex is at (6,-4).

Now you can find the side lengths and thus the perimeter.

To find the perimeter of the triangle, we need to determine the lengths of its sides.

Given the vertices (3, 0) and (6, 4), we can use the distance formula to find the length of the first side:

Side 1: Length from (3, 0) to (6, 4)

Using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Substituting the coordinates:
d = sqrt((6 - 3)^2 + (4 - 0)^2)
d = sqrt(3^2 + 4^2)
d = sqrt(9 + 16)
d = sqrt(25)
d = 5

So, the length of Side 1 is 5 units.

Next, we need to find the length of the other two sides. Since one of the angles is bisected by the x-axis, the triangle must have a right angle at one of its vertices. The side opposite to the right angle is the hypotenuse, which is usually the longest side of a right triangle.

To find the length of the hypotenuse (Side 2), we can use the Pythagorean Theorem:

Side 2: Length of the hypotenuse

Let's assume the third vertex of the triangle is (x, 0), where x represents the x-coordinate of that vertex. The length of Side 2 will be the distance between (6, 4) and (x, 0).

Using the distance formula:
d = sqrt((6 - x)^2 + (4 - 0)^2)

Since the angle bisector by the x-axis divides the triangle into two congruent right triangles, we can determine the x-coordinate of the third vertex as the midpoint between the x-coordinates of (3, 0) and (6, 4). That midpoint will also be the value of x that minimizes the distance between (6, 4) and (x, 0), making it the length of Side 2.

Midpoint formula:
x = (x1 + x2) / 2

Substituting the coordinates of (3, 0) and (6, 4):
x = (3 + 6) / 2
x = 4.5

Now we can calculate the length of Side 2:
d = sqrt((6 - 4.5)^2 + (4 - 0)^2)
d = sqrt(1.5^2 + 4^2)
d = sqrt(2.25 + 16)
d = sqrt(18.25)
d ≈ 4.27 (rounded to two decimal places)

So, the length of Side 2 is approximately 4.27 units.

Finally, to find the length of Side 3, we can subtract the lengths of Side 2 and Side 1 from the total length of the base (6 units or the difference between the x-coordinates of (3, 0) and (6, 4)):

Side 3: Length of the remaining side

d = 6 - 5 - 4.27
d ≈ 2.73 (rounded to two decimal places)

So, the length of Side 3 is approximately 2.73 units.

Now, we can find the perimeter of the triangle by adding up the lengths of all three sides:

Perimeter = Side 1 + Side 2 + Side 3
Perimeter ≈ 5 + 4.27 + 2.73
Perimeter ≈ 11 (rounded to two decimal places)

Therefore, the perimeter of the triangle is approximately 11 units.