Question

On a coordinate plane, triangle a b c has points (negative 9, 10), (7, negative 2) and (4, negative 6). the length of b c is 5. angle a b c is a right angle. triangle abc is a right triangle the length of bc is 5 units. the area of abc is square units.

Answer 1

The area of ∆ABC is 50 square units.

Explanation:

You want the area of right triangle ABC with coordinates A(-9, 10), B(7, -2), and C(4, -6).

Legs

The vector BC can be found as ...

C -B = (4, -6) -(7, -2) = (4 -7, -6 -(-2)) = (-3, -4)

The vector BA can be found as ...

A -B = (-9, 10) -(7, -2) = (-9 -7, 10 -(-2)) = (-16, 12)

Area

The area of the triangle will be half the absolute value of the "cross product" of these vectors:

1/2·|BC×BA| = 1/2|(-3(12) -(-4(-16))) = 1/2|-36 -64| = 50

The area of ∆ABC is 50 square units.

__

Additional comment

This "cross product" of the 2-dimensional vectors is the determinant of the 2×2 matrix of their components. This method of finding triangle area works with any triangles whose vertex coordinates are known.

`\left|\begin{array}{cc}a&b\\c&d\end{array}\right|=ad-bc`

<95141404393>