Question

triangle mno is congruent to right triangle rst with a right angle at vertex r. if the slope of rs is -1/5, what must be true?
the slope of tr is 5.
the slope of om is 5.
the slope of mn is -1/5
the slope of no is -1/5

Explanation:

Step1: Recall the property of congruent triangles

Congruent triangles have corresponding parts equal. Since \(\triangle MNO\cong\triangle RST\) and \(\angle R\) is a right - angle (so \(RS\perp RT\)), in \(\triangle MNO\), \(\angle M\) is the right - angle (corresponding to \(\angle R\)). The slope of a line perpendicular to a line with slope \(m\) is \(-\frac{1}{m}\) (if \(m
eq0\)). Given the slope of \(RS =-\frac{1}{5}\), the slope of \(TR\) (perpendicular to \(RS\)) should satisfy the perpendicular slope formula \(m_1\times m_2=- 1\). Let the slope of \(TR\) be \(m\), then \(-\frac{1}{5}\times m=-1\), so \(m = 5\). Also, in \(\triangle MNO\), \(MN\) corresponds to \(RS\) and \(NO\) corresponds to \(ST\) and \(OM\) corresponds to \(TR\).

Step2: Analyze each option

• For the slope of \(\overline{OM}\): Since \(\overline{OM}\) corresponds to \(\overline{TR}\) (because of the congruence \(\triangle MNO\cong\triangle RST\) and the right - angle correspondence), and we found the slope of the line perpendicular to \(RS\) (which is similar to the relationship between \(TR\) and \(RS\) in \(\triangle RST\) and \(OM\) and \(MN\) in \(\triangle MNO\)) has a slope of \(5\).
• For the slope of \(\overline{MN}\): \(\overline{MN}\) corresponds to \(\overline{RS}\), so its slope should be \(-\frac{1}{5}\), not \(-\frac{1}{5}\) for \(\overline{MN}\) (wrong correspondence assumption in the option).
• For the slope of \(\overline{NO}\): There is no basis from the congruence and slope - perpendicularity relationship to say its slope is \(-\frac{1}{5}\).

Answer:

The slope of \(\overline{OM}\) is \(5\)