The triangle has vertices A(0,1),B(1,6) and C(5,2).Prove that the triangle is isosceles.
Well we can graph the points and form a triangle. We can then find the distance between each point(side). And if two sides have the same distance we know that the triangle is isosceles.
Distance between AC:
D=square root of (5-0)^2 +(2-1)^2
D=radical 26
distance between AB:
D= square root (1-0)^2 +(6-1)^2
D=radical 26
Distance between BC:
D=square root (5-1)^2 +(2-6)^2
D=4 radical 2
Tada!
To prove that a triangle is isosceles, we need to show that at least two of its sides are equal in length. In this case, we will examine the lengths of the sides AB, BC, and AC.
Step 1: Calculate the lengths of the sides AB, BC, and AC.
To calculate the length of a side in a triangle, we use the distance formula, which is derived from the Pythagorean theorem. The distance formula is:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using the given coordinates, we can calculate the lengths of the sides AB, BC, and AC as follows:
AB = √((1 - 0)^2 + (6 - 1)^2)
BC = √((5 - 1)^2 + (2 - 6)^2)
AC = √((5 - 0)^2 + (2 - 1)^2)
Simplifying the equations, we get:
AB = √(1^2 + 5^2) = √(1 + 25) = √26
BC = √(4^2 + (-4)^2) = √(16 + 16) = √32
AC = √(5^2 + 1^2) = √(25 + 1) = √26
Step 2: Compare the lengths of the sides.
Now, we compare the lengths of the sides AB, BC, and AC.
Since AB = AC = √26, we see that two sides of the triangle are equal in length. Therefore, the triangle is isosceles.
In conclusion, using the distance formula to calculate the lengths of the sides AB, BC, and AC, we found that AB = AC = √26. This shows that the triangle is isosceles.