Question

Two right triangles are graphed on a coordinate plane. One triangle has a vertical side of 2 and a horizontal side of 6. The other triangle has a vertical side of 10 and a horizontal side of 30.
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> Could the hypotenuses of these two triangles lie along the same line?

Answer 1

Consider two right triangles:

1. ΔABC with vertices A(0,0), B(0,2), C(6,0). Then AB is perpendicular to AC and AB=2 units (vertical leg), AC=6 units (horizontal leg).

2. ΔXYZ with vertices X(6,-10), Y(6,0), Z(36,-10). Then XY is perpendicular to XZ and XY=10 units (vrrtical leg), XZ=30 units (horizontal leg).

The equation of the line BC is

$\dfrac{x-0}{6-0}=\dfrac{y-2}{0-2},\\ \\-x=3y-6,\\ \\x+3y-6=0.$

Check whether points Y and Z lie on this line:

Y(6,0): $6+3\cdot 0-6=0$ - true;

Z(36,-10): $36+3\cdot (-10)-6=0$ - true.

Answer:  the hypotenuses of these two triangles could lie along the same line