Question

The vertices of a triangle are S(-2, -2), T10,-2) and R(4, 4). Prove algebraically that this is a right - angledtriangle. (7A)

Answer 1

Right Triangle

If one of the interior angles of a triangle measures 90°, then we call the triangle a right triangle.

We must prove one of the angles formed by two of the sides of the triangle measures 90°. Those sides must be perpendicular to each other.

If two lines with slopes m1 and m2 are perpendicular, then:

m1 * m2 = -1

We'll calculate the slopes of each line defined by their endpoints (x1, y1) and (x2, y2) with the formula:

`m=(y_2-y_1)/(x_2-x_1)`

Calculate the slope of the segment ST:

`m_1_{}=(-2+2)/(10+2)=0`

This represents a horizontal line.

Now for segment TR:

`\begin{gathered} m_2=(4+2)/(4-10) \\ m_2=(6)/(-6)=-1 \end{gathered}`

For segment SR:

`m_3=(4+2)/(4+2)=1`

Note the product of m2 and m3 is -1, thus the triangle is right-angled at vertex R.