Show by vector calculation that the bisector of the angle between the two equal sides of an isosceles triangle bisects the third side and is perpendicular to it.
To show that the bisector of the angle between the two equal sides of an isosceles triangle bisects the third side and is perpendicular to it, we can use vector calculations.
Let's assume we have an isosceles triangle with vertices A, B, and C. The lengths of the equal sides are |AB| = |AC|, and the angle between them is θ. We want to show that the bisector of the angle θ, which we'll call BD, bisects the third side, BC, and is perpendicular to it.
Step 1: Define the vectors
We can represent the triangle sides as vectors. Let's define vector AB as vector u and vector AC as vector v. Also, let vector BD be vector x, and vector BC be vector y.
Step 2: Find the vector bisector
Since BD bisects the angle θ, we can represent vector BD as the sum of vector u and vector v, scaled such that its length is half the length of those vectors. Mathematically, we can express this as:
x = (u + v)/2
Step 3: Show that the bisector bisects the third side
To prove that the bisector bisects the third side, we need to show that vector y can be expressed as the sum of vectors x and v, both scaled by a factor of 1/2. Mathematically, we want to prove:
y = (x + v)/2
Step 4: Show that the bisector is perpendicular to the third side
To prove that the bisector is perpendicular to the third side, we need to show that the dot product of vectors x and y is zero. The dot product of two vectors can be calculated as:
(x ⋅ y) = |x| ⋅ |y| ⋅ cos(θ)
If we can show that the dot product is zero, it implies that the angle between x and y is 90 degrees, which means they are perpendicular.
Step 5: Perform the vector calculations
To complete the proof, calculate the vectors and their properties using the given information about the isosceles triangle. Substitute the values of vectors u and v into the expressions for x and y. Finally, calculate the dot product of vectors x and y. If the dot product is zero, it verifies that the bisector bisects the third side and is perpendicular to it.
By performing these vector calculations, you can demonstrate that the bisector of the angle between the two equal sides of an isosceles triangle indeed bisects the third side and is perpendicular to it.