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

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

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

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

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

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

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

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

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

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