4.5 The Converse of the Pythagorean Theorem Objective: Use the Converse of Pythagorean Theorem. Use side lengths to classify triangles.

Warm up: 1. What is the Pythagorean Theorem? 𝑎 2 + 𝑏 2 = 𝑐 2 where “c” is the longest side (hypotenuse) 2. When do we use the Pythagorean Theorem? Use in RIGHT triangles to find the side lengths

𝐼𝑓 𝑎 2 + 𝑏 2 = 𝑐 2 , 𝑡ℎ𝑒𝑛 ∆𝐴𝐵𝐶 𝑖𝑠 𝑎 𝒓𝒊𝒈𝒉𝒕 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒.

a= 12 b= 16 c= 20 longest side 𝑎 2 + 𝑏 2 ? 𝑐 2 (12) 2 + (16) 2 ? (20) 2 144 + 256 ? 400 400 ? 400 𝑎 2 + 𝑏 2 𝑑𝑜𝑒𝑠 𝑒𝑞𝑢𝑎𝑙 𝑐 2 , 𝑡ℎ𝑒𝑛 ∆𝐴𝐵𝐶 𝑖𝑠 𝑎 𝑅𝐼𝐺𝐻𝑇 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 400 = 400

Classifying Triangles: In ∆𝐴𝐵𝐶 with longest side c You can determine whether a triangle is acute, right, or obtuse by its side lengths.

a= 4 b= 5 c= 35 longest side 𝑎 2 + 𝑏 2 ? 𝑐 2 (4) 2 + (5) 2 ? ( 35 ) 2 16 + 25 ? 35 41 ? 35 𝑐 2 𝑖𝑠 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑎 2 + 𝑏 2 , 𝑡ℎ𝑒𝑛 ∆𝐴𝐵𝐶 𝑖𝑠 𝑎𝑛 𝐴𝐶𝑈𝑇𝐸 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 41 > 35

a= 8 b= 12 c= 15 longest side 𝑎 2 + 𝑏 2 ? 𝑐 2 (8) 2 + (12) 2 ? (15) 2 64 +144 ? 225 208 ? 225 𝑐 2 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑎 2 + 𝑏 2 , 𝑡ℎ𝑒𝑛 ∆𝐴𝐵𝐶 𝑖𝑠 𝑎𝑛 𝑂𝐵𝑇𝑈𝑆𝐸 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 208 < 225

a= 5 b= 6 c= 8 longest side 𝑎 2 + 𝑏 2 ? 𝑐 2 (5) 2 + (6) 2 ? (8) 2 25 +36 ? 64 61 ? 64 𝑐 2 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑎 2 + 𝑏 2 , 𝑡ℎ𝑒𝑛 ∆𝐴𝐵𝐶 𝑖𝑠 𝑎𝑛 𝑂𝐵𝑇𝑈𝑆𝐸 𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒 61 < 64

52 > 49 1369=1369 The triangle is acute. The triangle is right.

You Try! 29 < 36 289=289 98 >49 Obtuse Right Acute

625 >576 625=625 625 <676 Acute Right Obtuse

32 >25 180 <196 225=225 Acute Obtuse Right