Finding Triangle Vertex Value

We are given the vertices of a triangle $\Delta ABC$ as A(0, 0), B(a, 5), and C(-5, 5). The triangle is specified as right-angled at vertex A.

Vertices Coordinates

The coordinates of the vertices are:

• A = (0, 0)
• B = (a, 5)
• C = (-5, 5)

Right-angled Condition at A

For a triangle to be right-angled at vertex A, the sides AB and AC must be perpendicular. This means their dot product is zero. We first find the vectors representing these sides.

Vector Calculation

Vector AB is found by subtracting the coordinates of A from B:

$ \vec{AB} = B - A = (a - 0, 5 - 0) = (a, 5) $

Vector AC is found by subtracting the coordinates of A from C:

$ \vec{AC} = C - A = (-5 - 0, 5 - 0) = (-5, 5) $

Dot Product Condition

Since $\Delta ABC$ is right-angled at A, the dot product of vectors $\vec{AB}$ and $\vec{AC}$ must be 0:

$ \vec{AB} \cdot \vec{AC} = 0 $

Calculating the dot product:

$ (a)(-5) + (5)(5) = 0 $

$ -5a + 25 = 0 $

Solving for 'a'

Now, we solve the equation for 'a':

$ -5a = -25 $

$ a = \frac{-25}{-5} $

$ a = 5 $

Therefore, the value of 'a' is 5.