1. Section 6.4 Notes: Rectangles
EQ: What are the properties of the diagonals of a rectangle?

2. Diagonals of a Rectangle
Vocab!
Rectangle
Rectangle Corollary
Diagonals of a Rectangle
A parallelogram with four right angles
Opp. ∠’s are ≅
Consecutive ∠’s are supplementary
Diagonals bisect each other
If JKLM is a rectangle, then 𝐽𝐿 ≅ 𝐾𝑀 J K M L

3. Example 1
A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet, and LN = 6.5 feet, find KM.
𝐽𝐿 ≅ 𝐾𝑀
𝐽𝐿 = 𝐽𝑁 + 𝑁𝐿 where 𝐽𝑁 ≅ 𝑁𝐿
𝐽𝑁 =6.5 𝑠𝑜 𝐽𝐿 =13
Since 𝐽𝐿 ≅ 𝐾𝑀 , 𝐾𝑀 =13

4. Example 2
Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4) and m∠SUR = (3x – 2), solve for x.
𝑚∠𝑅𝑇𝑈+𝑚∠𝑆𝑈𝑅=90
8𝑥+4+3𝑥−2=90
11𝑥+2=90
𝑥=8

5. You Try!
Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ.
2. Quadrilateral EFGH is a rectangle. If m∠FGE = (6x – 5) and m∠HFE = (4x –5), solve for x.
𝐹𝐻 = 𝐺𝐸
𝐺𝐸 =15
15 2 =7.5
𝑚∠𝐹𝐺𝐸+𝑚∠𝐻𝐹𝐸=90°
6𝑥−5+4𝑥−5=90
10𝑥−10=90
𝑥=10

6. Converse of Diagonals of a Rectangle
If the diagonals of a parallelogram are ≅ then the parallelogram is a rectangle

7. Example 3
In rectangle QRST, QS= 5x – 31 and RT = 2x Find the lengths of the diagonals of QRST.
𝑅𝑇 ≅ 𝑄𝑆
5𝑥−31=2𝑥+11
𝑥=21
Diagonals = 5 21 −31=74

8. How to prove a rectangle with coordinate geometry?
By using the properties of rectangles, you can use the vertices to prove it’s a rectangle

9. Example 4
Quadrilateral JKLM has vertices J(–2, 3), K(1, 4), L(3, –2), and M(0, –3). Determine whether JKLM is a rectangle using the Distance Formula.
Opposite sides are congruent  prove this with the distance formula
𝐽𝐾 = ( 1−−2) 2 +( 4−3) 2 = 10
𝑀𝐿 = ( 3−0) 2 +( −2−−3) 2 = 10
𝐽𝑀 = ( 0−−2) 2 +( −3−3) 2 = 40
𝐾𝐿 = ( 3−1) 2 +( −2−4) 2 = 40

10. You Try!
Graph the quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula.
A(–3, 1), B(–3, 3), C(3, 3), D(3, 1)
𝐴𝐵 =2 𝑢𝑛𝑖𝑡𝑠
𝐶𝐷 =2 𝑢𝑛𝑖𝑡𝑠
𝐶𝐵 =6 𝑢𝑛𝑖𝑡𝑠
𝐴𝐷 =6 𝑢𝑛𝑖𝑡𝑠
Opposite sides are congruent so therefore this quadrilateral is a rectangle. B C A D