# TODAY IN GEOMETRY… Warm Up: Simplifying Radicals

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TODAY IN GEOMETRY… Warm Up: Simplifying Radicals Practice: Solving missing sides using the Pythagorean Theorem Learning Target : Use the Converse of the Pythagorean Theorem determine if a triangle is a right triangle Independent Practice

Perfect Squares: 1 4 9 16 25 36 49 64 81 100 121 144 WARMUP: MULTIPLYING AND SIMPLIFYING RADICALS: 1. ( 5 ) 2 + ( 6 ) · 20 ( 3 ) 2 · (2 6 ) 2 6. (8 12 ) 2 =4 20 =5+6 =4 4·5 =4( 4 · 5 ) =4(2 5 ) =𝟖 𝟓 =𝟏𝟏 =10(3) =𝟑𝟎 = 2 2 · =4(6) =𝟐𝟒 = 4·7 8·3 =28( 24 ) = · 6 =28(2 6 ) =𝟓𝟔 𝟔 = 8 2 · 12 2 =64(12) =𝟕𝟔𝟖

USING THE PYTHAGOREAN THEOREM TO FIND MISSING SIDES
REVIEW: USING THE PYTHAGOREAN THEOREM TO FIND MISSING SIDES

PRACTICE: Identify the unknown side as a leg or hypotenuse
PRACTICE: Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle. Write your answer in the simplest radical form. The unknown side is not attached to the right angle so it is a hypotenuse of the triangle. Use the Pythagorean theorem to find the missing leg: (𝑙𝑒𝑔) 2 + (𝑙𝑒𝑔) 2 = (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒) 2 (17) 2 + (32) 2 = 𝑥 2 = 𝑥 2 1313= 𝑥 2 1313 = 𝑥 2 𝟏𝟑𝟏𝟑 =𝒙≈𝟑𝟔.𝟐 𝑥 17 32

PRACTICE: Identify the unknown side as a leg or hypotenuse
PRACTICE: Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle. Write your answer in the simplest radical form. The unknown side is attached to the right angle so it is a leg of the triangle. Use the Pythagorean theorem to find the missing leg: (𝑙𝑒𝑔) 2 + (𝑙𝑒𝑔) 2 = (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒) 2 (26) 2 + 𝑥 2 = (58) 2 676+ 𝑥 2 =3364 − − 676 𝑥 2 =2688 𝑥 2 = 2688 𝒙=𝟖 𝟒𝟐 ≈𝟓𝟏.𝟖 26 𝑥 58

PRACTICE: Identify the unknown side as a leg or hypotenuse
PRACTICE: Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle. Write your answer in the simplest radical form. 6 8 12 𝑥 The unknown side is not attached to the right angle so it is a hypotenuse of the triangle. (There are two right triangles. You must find a before you can find x.) Use the Pythagorean theorem to find a: (𝑙𝑒𝑔) 2 + (𝑙𝑒𝑔) 2 = (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒) 2 (6) 2 + (8) 2 = 𝑎 2 36+64= 𝑎 2 100= 𝑎 2 100 = 𝑎 2 𝟏𝟎=𝒂 𝑎 =10 Use the Pythagorean theorem again to find x: (𝑙𝑒𝑔) 2 + (𝑙𝑒𝑔) 2 = (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒) 2 (10) 2 + (12) 2 = 𝑥 2 = 𝑥 2 244= 𝑥 2 244 = 𝑥 2 2 61 =𝒙≈𝟏𝟓.𝟔

CONVERSE TO THE PYTHAGOREAN THEOREM:
3 PARTS!!!

CONVERSE TO THE PYTHAGOREAN THEOREM:
𝑎 𝑐 𝑏 𝐵 𝐴 𝐶 PART 1: RIGHT TRIANGLE if… PART 2: ACUTE TRIANGLE if… PART 3: OBTUSE TRIANGLE if… 𝑎 2 + 𝑏 2 = 𝑐 2 𝑎 2 + 𝑏 2 > 𝑐 2 𝑎 2 + 𝑏 2 < 𝑐 2

PRACTICE: Tell whether a triangle with the given side lengths is a right triangle.
Assign the given lengths to the sides of the triangle. The largest side is always assigned the hypotenuse! Use the Pythagorean theorem to check if it is a right triangle. (𝑙𝑒𝑔) 2 + (𝑙𝑒𝑔) 2 = (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒) 2 (4) 2 + (8) 2 = ? 16+64=1849 ? 80=1849 ? 𝟖𝟎≠𝟏𝟖𝟒𝟗 NOT A RIGHT ANGLE! 80<1849 TRIANGLE IS OBTUSE! 4, 43, 8 43 4 8 4 8 43

PRACTICE: Tell whether a triangle with the given side lengths is a right triangle.
Assign the given lengths to the sides of the triangle. The largest side is always assigned the hypotenuse! Use the Pythagorean theorem to check if it is a right triangle. (𝑙𝑒𝑔) 2 + (𝑙𝑒𝑔) 2 = (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒) 2 (10) ) 2 = ? =196 ? 221=196 ? 𝟐𝟐𝟏≠𝟏𝟗𝟔 NOT A RIGHT ANGLE! 221>196 TRIANGLE IS ACUTE! 10, 11, 14 14 10 11 11 10 14

PRACTICE: What kind of triangle (acute, obtuse or right) can be formed with segment lengths 4.3, 5.2, 6.1 Assign the given lengths to the sides of the triangle. The largest side is always assigned the hypotenuse! Use the Pythagorean theorem to check if it is a right triangle. (𝑙𝑒𝑔) 2 + (𝑙𝑒𝑔) 2 = (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒) 2 (4.3) 2 + (5.2) 2 = (6.1) 2 ? =37.21 ? 45.53=37.21 ? 𝟒𝟓.𝟓𝟑≠𝟑𝟕.𝟐𝟏 NOT A RIGHT ANGLE! 45.53>37.21 TRIANGLE IS ACUTE! 6.1 4.3 5.2 4.3 5.2 6.1

PRACTICE: Graph points A, B and C. Connect the points to form △𝐴𝐵𝐶
PRACTICE: Graph points A, B and C. Connect the points to form △𝐴𝐵𝐶. Decide whether the triangle is acute, right or obtuse. 𝐴 −2, 4 𝐵 6, 0 𝐶(−5, −2) Use the distance formula to find the segment lengths: 𝑑= ( 𝑥 2 − 𝑥 1 ) 2 + ( 𝑦 2 − 𝑦 1 ) 2 𝐴𝐵= (6− −2 ) 2 + (0−4) 2 = 80 𝐵𝐶= (−5−6) 2 + (−2−0) 2 = 125 𝐶𝐴= (−5− −2 ) 2 + ( −2 −4) 2 = 45 Use the Pythagorean theorem to check if it is a right triangle. (𝑙𝑒𝑔) 2 + (𝑙𝑒𝑔) 2 = (ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒) 2 ( 45 ) 2 + ( 80 ) 2 = ( 125 ) 2 ? 45+80=125 ? 125=125 ? 𝟏𝟐𝟓=𝟏𝟐𝟓 IT IS A RIGHT TRIANGLE! 𝑨 𝑩 𝑪

HOMEWORK #2: Pg. 444:1-6, 8, 10, 12 If finished, work on other assignments: HW #1: Pg. 436: 3-29 odd