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:

1. Use the distance formula to measure the lengths of the sides: The distance formula ($d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$) can find side lengths, which helps classify triangles (e.g., equilateral, isosceles, scalene) and quadrilaterals (e.g., by side - length properties). So this applies.
2. Use the slope to determine whether opposite sides are parallel: Parallel lines have equal slopes. For quadrilaterals, checking if opposite sides have equal slopes tells us if they are parallel (e.g., in a parallelogram, opposite sides are parallel). This is useful for classification, so this applies.
3. Use the slope to check whether sides are perpendicular and form right angles: Perpendicular lines have slopes that are negative reciprocals ($m_1\times m_2=- 1$). For triangles, this can identify right - angled triangles, and for quadrilaterals (e.g., rectangles have right angles), so this applies.
4. Use the distance formula to compare whether opposite sides are congruent: Congruent sides have equal lengths. Using the distance formula to find side lengths and then comparing opposite sides helps classify quadrilaterals (e.g., in a parallelogram, opposite sides are congruent), so this applies.
5. Use the slope to check whether the diagonals are perpendicular to each other: For quadrilaterals like rhombuses, diagonals are perpendicular. Checking the slopes of diagonals (using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$) to see if they are negative reciprocals helps in classification, so this applies.
6. Use the distance formula to compare whether diagonals are congruent: For quadrilaterals like rectangles, diagonals are congruent. Using the distance formula to find the lengths of diagonals and comparing them helps in classification, so this applies.

Answer:

• 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.