an isosceles triangle on the coordinate plane has side lengths of 8,5,and 5 units. The long side is along the x-acis, find the coordinates of each vertex.
place the long side on the x-axis so that it runs from
(0,0) to (8,0)
then the altitude cuts the base in half and hits it at (4,0)
looks like we have 2 right-angled triangles with base of 4 and hypotenuse of 5,
by Pythagoras that height has to be 3
(s^2 + 4^2 = 5^2)
so the third vertex must be (4,3)
To find the coordinates of each vertex of the isosceles triangle on the coordinate plane, we can use a few steps.
Step 1: Determine the location of one vertex.
Since the longest side of the triangle is along the x-axis, one vertex will be located at (x, 0), where x is the x-coordinate of the vertex we are trying to determine.
Step 2: Use the property of isosceles triangles.
The isosceles triangle has two sides of equal length (5 units in this case). This means that the other two vertices will be equidistant from the first vertex.
Step 3: Determine the remaining vertices.
Given that the side lengths are 8, 5, and 5 units, we can deduce that the other two vertices will each be 5 units away from the first vertex.
Since one vertex is located at (x, 0), the other two vertices could be located at (x - 5, y) and (x + 5, y), where y represents the y-coordinate of both vertices.
Step 4: Find the value of x.
To solve for x, we should consider the fact that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, the sum of the two sides with length 5 units is 10, which is equal to the length of the side with length 8 units.
Therefore, we can conclude that the x-coordinate of the first vertex is 8/2 = 4.
Step 5: Find the y-coordinate.
Since we now know that the x-coordinate of the first vertex is 4, we can determine the y-coordinate by using the Pythagorean theorem. Considering the right triangle formed by the vertices (4, 0), (4-5, y), and (4+5, y), we can solve for y.
The distance between the two vertex points on the y-axis is given as 8 units (5 units on each side plus the remaining unit for the x-coordinate). This represents the hypotenuse of the right triangle.
Applying the Pythagorean theorem, we have:
(8)^2 = 5^2 + (2y)^2
Simplifying the equation:
64 = 25 + 4y^2
39 = 4y^2
y^2 = 9.75
y ≈ ±3.12
Therefore, the other two vertices of the isosceles triangle are approximately (4-5, 3.12) ≈ (-1, 3.12) and (4+5, 3.12) ≈ (9, 3.12).
Hence, the coordinates of the vertices of the isosceles triangle are (4, 0), (-1, 3.12), and (9, 3.12).