It is clear that the vertex (2,a) is the right angle vertex and the segment [(0,0 - (10,0)] is the hypotenuse.

Then you can write the Pythagorean equation in the form

 = .       (1)

It is your equation to find the unknown value of "a".

Simplify and solve it:

 = ,

 = ,

 =   --->   =   --->  a = 4.

Now, the hypotenuse has the length 10, while the altitude drawn to the hypotenuse has the length 4.

Hence, the area of the triangle is  = 20 square units.

Answer.  The area of the triangle is 20 square units.

Let A be the point (0,0) in the coordinate plane (the origin);
let B be the point (2,a), and let C be the point (10,0).
Then the triangle ABC is right-angled triangle, according to the condition.
with the vertex B as the right angle and the segment AC as the hypotenuse.

Next, draw the perpendicular from the vertex B to the hypotenuse AC. The foot of this perpendicular is the point D = (2,0).

I recommend you to make a sketch of the triangle ABC and draw this perpendicular CD.

Now you have two right angled triangles, ADB and CDB.

From the triangle ADB you have  = .
From the triangle CDB you have  = .

The equation (1) in my very first post is

 +  = .

Everything is so obvious . . .