Math  /  Geometry

QuestionGraph the points A(2,3),B(7,8),C(9,6)A(2,-3), B(7,-8), C(9,-6), and D(4,1)D(4,-1). Draw rectangle ABCDA B C D and diagonals AC\overline{A C} and BD\overline{B D}. a. Find the midpoints of AC\overline{A C} and BD\overline{B D}. b. What appears to be true about the diagonals of a rectangle? a. The midpoint of AC\overline{\mathrm{AC}} is (112,92)\left(\frac{11}{2}, \frac{-9}{2}\right). (Type an ordered pair.)
The midpoint of BD\overline{B D} is \square . (Type an ordered pair.) Clear all

Studdy Solution

STEP 1

1. The points A(2,3) A(2, -3) , B(7,8) B(7, -8) , C(9,6) C(9, -6) , and D(4,1) D(4, -1) are vertices of a rectangle.
2. The diagonals of a rectangle bisect each other.

STEP 2

1. Plot the points and draw the rectangle and its diagonals.
2. Calculate the midpoint of diagonal AC\overline{A C}.
3. Calculate the midpoint of diagonal BD\overline{B D}.
4. Analyze the midpoints to determine the property of the diagonals.

STEP 3

Plot the points A(2,3) A(2, -3) , B(7,8) B(7, -8) , C(9,6) C(9, -6) , and D(4,1) D(4, -1) on a coordinate plane. Connect the points in the order ABCDA A \to B \to C \to D \to A to form rectangle ABCD ABCD . Draw the diagonals AC\overline{A C} and BD\overline{B D}.

STEP 4

Calculate the midpoint of diagonal AC\overline{A C} using the midpoint formula:
MAC=(x1+x22,y1+y22) M_{AC} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
Substitute the coordinates of points A(2,3) A(2, -3) and C(9,6) C(9, -6) :
MAC=(2+92,3+(6)2) M_{AC} = \left( \frac{2 + 9}{2}, \frac{-3 + (-6)}{2} \right) =(112,92) = \left( \frac{11}{2}, \frac{-9}{2} \right)

STEP 5

Calculate the midpoint of diagonal BD\overline{B D} using the midpoint formula:
MBD=(x1+x22,y1+y22) M_{BD} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
Substitute the coordinates of points B(7,8) B(7, -8) and D(4,1) D(4, -1) :
MBD=(7+42,8+(1)2) M_{BD} = \left( \frac{7 + 4}{2}, \frac{-8 + (-1)}{2} \right) =(112,92) = \left( \frac{11}{2}, \frac{-9}{2} \right)

STEP 6

Analyze the midpoints calculated in steps 2 and 3. Since both midpoints are the same, it confirms that the diagonals of rectangle ABCD ABCD bisect each other.
The midpoint of BD\overline{B D} is (112,92)\boxed{\left( \frac{11}{2}, \frac{-9}{2} \right)}.

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