CONFIDENTIAL 1 Grade 8 Pre-Algebra Pythagorean Theorem 2

Pythagorean Theorem The Pythagorean Theorem shows how the legs and hypotenuse of a right triangle are related. legs hypotenuse In a right triangle, the two shortest sides are legs. The longest side, which is opposite the right angle, is the hypotenuse.

In words: In a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the legs. In Symbols: a 2 +b 2 = c 2. a b c If we know the lengths of two sides of a right angled triangle, then Pythagoras' Theorem allows us to find the length of the third side. Pythagorean Theorem

4 hypotenuse = 2. shorter leg longer leg = shorter leg. √3 C 30º 60º 2s A B s √3s Using the Pythagorean theorem, we find that The converse of Pythagorean Theorem allows you to substitute the lengths of the sides of a triangle into the equation : c 2 = a 2 +b 2 to check whether a triangle is a right triangle, if the Pythagorean equation is true the triangle is a right triangle.

Let us do some practice problems!

CONFIDENTIAL 6 1) Is every isosceles triangle an isosceles right triangle? a)True b)False

CONFIDENTIAL 7 2) A ladder that is 10 feet long leans against a building. The bottom of the ladder is 4 feet away from the base. How far up the side of the building does the ladder reach? a)7.2 ft b)10.2 ft c)8.2 ft d)9.2 ft

CONFIDENTIAL 8 3) If one side of a right triangle is 3 ft and the length of its hypotenuse is 4 ft, then find the length of the other side. a)√3 ft b)√5 ft c)√7 ft d)√2 ft c) √7 ft

CONFIDENTIAL 9 4) A ladder which is 25 ft long is placed on a wall such that the top of the ladder touches the top of the wall. The bottom of the ladder is 15 ft away from the wall. What is the height of the wall? a)20 ft b)15 ft c)10 ft d)16 ft a) 20 ft

CONFIDENTIAL 10 5) If a = 12 cm and b = 14 cm, then find the value of c in the right triangle. a)15.63 cm b)14.63 cm c)18.43 cm d)12.43 cm c) cm

CONFIDENTIAL 11 6) The vertices of 3 squares are joined to form a right triangle. What is the area of the smallest square? 54 cm 2 36 cm 2 1)18 cm 2 2)20 cm 2 3)22 cm 2 4)14 cm 2 a) 18 cm 2

CONFIDENTIAL 12 7) The floor mat used in gymnastics measures 40 feet by 40 feet. Gymnasts use the diagonal of the mat for most of their tumbling patterns. How many feet are available to the gymnasts by using the diagonal of the mat for their tumbling patterns. a)49.2 ft b)50.2 ft c)66.6 ft d)56.6 ft

CONFIDENTIAL 13 8) The length and the width of a rectangular football court are 40 feet and 45 feet respectively. A diagonal walkway is made from one end to the opposite end. What is the length of the walkway? a)61.2 ft b)60.2 ft c)59.2 ft d)None of the above b) 60.2 ft

CONFIDENTIAL 14 9) The lengths of the perpendicular sides of a right angled triangle 3 cm. and 4 cm. Then find the perimeter of the triangle. d) 12 cm a)18 cm b)16 cm c)14 cm d)12 cm

CONFIDENTIAL 15 10) One leg of a right triangle is 15 inches long. The other leg is 3 inches less than the hypotenuse. What are the lengths of this leg and hypotenuse? a)27 in b)34 in c)36 in d)35 in c) 36 in

Assessment

CONFIDENTIAL 17 1) Figure ABCD is a rectangular field 136 ft long and 102 ft wide. By how much is the path from A to C shorter than the path from A to B to C? a)68 ft b)66 ft c)72 ft d)70 ft a) 68 ft

CONFIDENTIAL 18 2) The length and the width of a rectangular field are 28 feet and 21 feet respectively. A diagonal walkway is made from one end to the opposite end. What is the length of the walkway? a)39 ft b)37 ft c)35 ft d)None of the above c) 35 ft

CONFIDENTIAL 19 3) State which of the following measures will form a right triangle. a)12, 14, 20 b)24, 18, 30 c)12, 13, 15 d)None of the above b) 24, 18, 30

CONFIDENTIAL 20 4) The perimeter of a right triangle is 150 inches and the ratio of the lengths of the 2 legs is 5 : 12. Find the length of the hypotenuse. a)70 in b)75 in c)60 in d)65 in

CONFIDENTIAL 21 5) Sheath went to a level field to fly a kite. She let out all 85 ft of string and tied it to a stake. Then she walked out on the field until she was directly under the kite, 51 ft from the stake. How high was the kite? a)68 ft b)70 ft c)72 ft d)66 ft a) 68 ft

CONFIDENTIAL 22 6) What is the length of the third side of the triangle shown in the figure? [Given a = 6.6 in. and b = 8.8 in..] 6.6 in 8.8 in x a)12 in b)10 in c)11 in d)13 in c) 11 in

CONFIDENTIAL 23 7) What is the measure of the side of the square, if the diagonal of the square is 8 feet? a)3.7 ft b)5.7 ft c)4.7 ft d)6.7 ft b) 5.7 ft

CONFIDENTIAL 24 8) The lengths of the perpendicular sides of a right angled triangle 30 cm. and 16 cm. Then find the perimeter of the triangle. a)75 cm b)72 cm c)78 cm d)80 cm

CONFIDENTIAL 25 9) If one side of a right triangle is two times the other and the length of hypotenuse is 25 ft, then what are the measures of the two sides? a)5√5 ft b)4√5 ft c)7√5 ft d)6√5 ft a) 5√5 ft

CONFIDENTIAL 26 10) The lengths of the hypotenuse and the shorter leg of a right triangle are in the ratio 5:3 and the perimeter of the triangle is 24 inches. Find the length of the longer leg. a)7 in b)8 in c)8 in d)8 in b) 8 in