Question

4 multiple answer 25 points
how can you show that the figure below is a rectangle? check all that apply.
graph of quadrilateral fjhg on a coordinate grid with axes x and y, origin o

• show that there are four right angles by using the slope formula.
• show the diagonals are perpendicular by using the slope formula.
• show the diagonals are congruent by using the distance formula.
• show that opposite sides are congruent by using the slope formula

Explanation:

Step1: Recall Rectangle Properties

A rectangle has four right angles (adjacent sides perpendicular), and diagonals are congruent. Also, opposite sides are congruent and parallel.

Step2: Analyze Each Option

• Option 1 (Four right angles via slope): If two lines are perpendicular, their slopes multiply to -1. Using slope formula to check adjacent sides' slopes (product = -1) shows right angles. Valid for rectangle.
• Option 2 (Diagonals perpendicular): Diagonals of a rectangle are congruent, not necessarily perpendicular (that's a rhombus/ square). So this is incorrect.
• Option 3 (Diagonals congruent via distance): A rectangle’s diagonals are congruent. Using distance formula (\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)) to check diagonals’ lengths works. Valid.
• Option 4 (Opposite sides congruent via slope): Slope formula finds steepness, not length. To check congruence, use distance formula. So this is incorrect.

Answer:

• Show that there are four right angles by using the slope formula.
• Show the diagonals are congruent by using the distance formula.