Question

Triangle DEF is congruent to right triangle GHI with a right angle at vertex H. If the slope of DE is –2, what must be true?

Answer 1

The slope of HI is 1/2

The slope of EF is 1/2

Solution:

Given that,

Triangle DEF is congruent to right triangle GHI

Which means,

These pairs of angles are congruent

{D, G}, {E, H}, and {F, I}

In triangle DEF, E is a right angle

This means that the line segments $\overline{DE}\ and\ \overline{EF}$ are perpendicular.

We know that,

Product of slope of a line and slope of line perpendicular to that line is equal to -1

Given that,

Slope of DE = -2

$\text{ Slope of DE } \times \text{ slope of EF} = -1\\\\-2 \times \text{ slope of EF} = -1\\\\\text{ slope of EF} = \frac{1}{2}$

Since the sides EF and HI are congruent,

Slopes of parallel lines are equal

$Slope\ of\ HI\ = \frac{1}{2}$

Thus, Slope of HI is 1/2