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Pythagorean Theorem Name_______________________ Day 1 Period______Date_____________ If an altitude AC is dropped from the right angle in a right triangle, three triangles are formed. What’s true about these 3 triangles and why? Remembering that corresponding sides of similar triangles are proportional, three extended proportions can be written. Write these below. The geometric mean between a and b is a number x so that a x = . There are 3 geometric mean x b statements in the extended proportions above. Write those geometric means below. Use this information to solve the following problems: 1. If CD = 9 and BC = 4, find AC.__________ Then find AD. __________ 2. If BC = 2 and BD = 8, find AB.__________ Then find AC.___________ 3. If CD = 6 and BC = 3, find AD.__________ Then find AB.___________ In each of the above problems, you could use the Pythagorean Theorem to work the last part. That’s because one proof of the Pythagorean Theorem uses this geometric mean. Given: h is the altitude to the hypotenuse of right triangle ABC. Prove: a2 + b2 = c2 1. h is the altitude to the hypotenuse of right triangle ABC. 1. Given 2. ADC and BDC are right angles 2. Altitudes form right angles. 3. ACD BDC ACB 3. All right angles are congruent. 4. A A and B B 4. 5. ∆ABC ~ ∆ACD and ∆ABC ~∆CBD 5. 6. y a x b = and = b c a c 7. b2 = x∙c and a2 = y∙c 6. 7. 2 2 8. 2 2 9. a + b = c∙(x + y) 9. 10. x + y = c 10. 11. a2 + b2 = c∙c = c2 11. 8. a + b = x∙c + y∙c Actually there are 364 proofs of the Pythagorean Theorem. Find another one of them and bring it in tomorrow! If ∆ PQR is a right triangle with right angle Q, solve each of the following problems: 1. If p = 5 and r = 7, find q._______________ 2. If p = 8 and r = 6, find q._______________ 3. If p = 3 and q = 8, find r. _______________ 4. If r = 5 and q = 13, find p.______________ Problems 2 and 4 are special because the right triangles in those problems had integer values for all 3 sides. The numbers {6, 8, 10} and {5, 12, 13} are called Pythagorean triples because they are integer values so that a2 + b2 = c2. Pythagorean triples and similarity relationships are often found on standardized tests. There are many more Pythagorean triples. Write down any odd number. ________ Square that odd number and divide it by 2. _________. Round this number up _______ and round this number down. ________ The 3 integers will form a Pythagorean triple. Prove that below. {3, 4, 5}, {5, 12, 13}, {7, 24, 25} and {8, 15, 17} are the most common Pythagorean triples. Knowing these and similar triangle properties can make finding sides of right triangles easier. Consider the following: Two legs of a right triangle are 30 7 and 40 7 . What is the hypotenuse? Solution: The common factor of these 2 numbers is 10 7 , and if we factor out that common factor, we get that legs of a similar triangle are 3 and 4. The hypotenuse of a triangle with legs of 3 and 4 is 5, so the hypotenuse of our triangle is 5(10 7 ) = 50 7 . Try solving these: 1. Two legs of a right triangle are 15 and 36. Find the hypotenuse.___________ 2. The hypotenuse of a right triangle is 100. If one leg has length 28, find the other leg._______ The distance formula is based on the Pythagorean Theorem too. Plot (2, 5) and (-1, 1). Draw a right triangle on this grid with these points as the endpoints of the hypotenuse. Show how the Pythagorean Theorem is used to find the distance between these points. Pythagorean Theorem Name_______________________ Day 2 Period______Date_____________ Review: In the figure to the right, b = 60 and c = 156. Find each of the following: a = ___________ d = ______________ e = ___________ f = ______________ As you saw yesterday, some right triangles have all integral side lengths these side lengths are called Pythagorean Triples. Other triangles are special because of their angles. Consider the following situation: AM is the altitude of EQUILATERAL triangle ABC. Construct that triangle below and altitude AM. What is mABM?________, mAMB________ Let the side of your triangle be any even Number. ___________ Why is MC = MB? What is MC?__________ Use the Pythagorean Theorem to find AM. B C ∆ABM is called a 30°-60°-90° triangle. Consider the equilateral triangle below with side length 2s. The altitude will create 2 30°-60°-90° triangles. Clearly, the shorter leg of the 30°-60°-90° triangle will be half the length of the hypotenuse. Show work to find the length of the altitude, which is the longer leg of the 30°-60°-90° triangle. The longer leg of a 30°-60°-90°is 3 times the length of the shorter leg. Try these: 1. If the shorter leg of a 30°-60°-90° triangle is 17, find the hypotenuse:_________ and the longer leg.___________ 2. If the hypotenuse of a 30°-60°-90° triangle is 100, find the shorter leg: ________ and the longer leg.___________ 3. If the longer leg of a 30°-60°-90° triangle is 6, find the shorter leg:_________ and the hypotenuse._____________ Pythagorean Theorem Name_______________________ Day 3 Period______Date_____________ Review: In the figure to the right, b = 60 and mA = 60. Find each of the following: d = ___________ f = ______________ e = ___________ a = ______________ c = ___________ State why each of the following pairs of triangles are congruent: ____________ _______________ _________________ ____________ ΔABC ΔEDC ΔABE ΔACD ΔADT ΔEDT ΔPRO ΔPST P R O T S You know triangles cannot be proven congruent using “SSA” because that theorem does not exist. HOWEVER, consider the two RIGHT triangles ΔABC and ΔXZY. If C and Y are right angles, legs, AC = ZZ = 15 and hypotenuses AB XZ = 17, that is an SSA situation. But – aren’t these triangles congruent? What makes this situation different is that in a right triangle, if we know the lengths of 2 sides, we know the length of the third side. That means the SSA case works IF the triangle is a right triangle. However, since it is a special case, we call this the HL theorem, which means “hypotenuseleg”. And it reads: If there exists a correspondence between the vertices of two right triangles such that the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other triangle, then the two right triangles are congruent. (HL) Consider the following proof: D Given: AB BC, DB is an altitude Prove: DB is an angle bisector A B C 1. AB BC, DB is an altitude 1. Given 2. DB AC 2.___________________________ 3. ABD and CBD are right angles. 3. ___________________________ 4. BD BD 4. ___________________________ 5. ΔABD ΔCBD 5. ___________________________ 6. BDA BDC 6. ___________________________ 7. DB is an angle bisector. 7. ___________________________ More fun with right triangles!! You will be assigned an integer. Write that integer here:__________ Show work to find the length of the diagonal of a square with the side length written above. You used a right triangle to solve this problem. What are the angles in this triangle? Compare notes with your partner. Can you generalize the situation? This triangle is called a 45-45-90 triangle and this is another special right triangle because of its angles. So, in conclusion: In any 30°-60°-90° triangle, the length of the hypotenuse is twice the length of the shorter leg and the length of the longer leg is 3 times the length of the shorter leg. and In any 45°-45°-90° triangle, the legs are congruent and the hypotenuse is length of a leg. 2 times the