Question

make a conjecture. how could the distance formula and slope be used to classify triangles and quadrilaterals in the coordinate plane? check all that apply. use the distance formula to measure the lengths of the sides. use the slope to determine whether opposite sides are parallel. use the slope to check whether sides are perpendicular and form right angles. use the distance formula to compare whether opposite sides are congruent. use the slope to check whether the diagonals are perpendicular to each other. use the distance formula to compare whether diagonals are congruent.

Explanation:

• The distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) can be used to find the lengths of sides (so we can check for congruent sides, whether sides are equal in length). For example, in a quadrilateral, if \(AB=\sqrt{(x_{A}-x_{B})^2+(y_{A}-y_{B})^2}\) and \(CD=\sqrt{(x_{C}-x_{D})^2+(y_{C}-y_{D})^2}\), we can compare \(AB\) and \(CD\) to see if they are congruent. Also, for triangles, we can check if it is isosceles (two sides equal) or equilateral (three sides equal).
• The slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). If two lines (sides) have the same slope \(m_1=m_2\), they are parallel. If the product of the slopes of two lines \(m_1\times m_2=- 1\), the lines are perpendicular. For example, in a quadrilateral, if \(m_{AB}=m_{CD}\) and \(m_{BC}=m_{AD}\), it may be a parallelogram. If \(m_{AB}\times m_{BC}=-1\), there is a right - angle at the vertex where \(AB\) and \(BC\) meet. For diagonals of a quadrilateral, if \(m_{diagonal1}\times m_{diagonal2}=-1\), the diagonals are perpendicular. And if \(d_{diagonal1}=d_{diagonal2}\) (using the distance formula for the endpoints of the diagonals), the diagonals are congruent.

Answer:

All of the options apply.

• Use the distance formula to measure the lengths of the sides.
• Use the slope to determine whether opposite sides are parallel.
• Use the slope to check whether sides are perpendicular and form right angles.
• Use the distance formula to compare whether opposite sides are congruent.
• Use the slope to check whether the diagonals are perpendicular to each other.
• Use the distance formula to compare whether diagonals are congruent.