# Pythagorean Theorem Problems

Problem 1 The distances in straight line from Isperih to Tutrakan and to Dulovo are 40km and 28km respectively, conecting the three cities they make right angle at Isperih. Find the distance from Dulovo to Tutrakan.

Solution If the required distance is x, then x2 = 402 + 282 = 1600 + 784 = 2384, x2 = 2384 => x = √2384 ≈ 50km.

Problem 2 Prove that the triangle with sides 3cm, 4cm and 5cm is right-angled triangle.

Solution 32 + 42 = 9 + 16 = 25 = 52. Becouse of 32 + 42 = 52, the triangle with these lengths of the sides is right-angled (with hypothenuse 5cm).

Problem 3 In the table belowwe have the distances between the points, A, B and C. Check if the points are points of right-angled triangle.

 AB BC AC a) 9 12 15 b) 5 11 12 c) 8,8 11,7 14,8 d) 5 9 8 e) 41 9 40 f) 6 4,5 7,5

b) - no;
c) - yes;
d) - no;
e) - yes;
f) - yes.

Problem 4 Prove if a triangle is right-angled if the lengths of its sides are:
a) √2, 4, 3√2;
b) √3, 3, 2√3;
c) √3, √5, √8.
b) - yes;
c) - yes.

Problem 5 Define the length of the hypothenuse of right-angled triangle, if hte lengths of the other two sides are:
a)√2 and √3;
b) √5 and √7;
c) √9 and √11.
b) - √12;
c) - √20.

Problem 6 In the parallelogram ABCD AB = 33cm, BC = 56cm and AC = 65cm. Check if the parallelogram is rectangular.

Problem 7 The mediums of sides of rhombus are points of a rectagular. If the lengths of the sides of the rectangular are à and b, what is the length of the side of the rhombus.

Problem 8 The side of a rectangular are 10cm and 24cm. Find the radius of the perscribed circumference aroud the recngular.

Problem 9 The hypothenuse of a right-angled triangle is 10cm, and one of its other sides is - 8cm. Find the face of the triangle.