# Pythagorean Theorem - Problems

**Problem 1**

The distance between town A and B is 40 miles, between B and C is 28 miles.
The three towns form a right angle at B. Find the distance between town A and town B.

**Solution:** Let denote the unkown distance be x. Then x^{2} = 40^{2} + 28^{2} = 1600 + 784 = 2384

x^{2} = 2384

$x = \sqrt{2384} \approx 49$ miles.

**Problem 2**

Prove that the triangle is right if its sides measure 3in, 4in and 5in.

**Solution** 3^{2} + 4^{2} = 9 + 16 = 25 = 5^{2}.
According the Pythagorean theorem the triangle is right.

**Problem 3**

In the table below are given distances between points A, B and C. Check if the points form a right 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 |

Answer:a) - yes;

b) - no;

c) - yes;

d) - no;

e) - yes;

f) - yes.

**Problem 4**

Prove that if a triangle is right, the lengths of its sides can be:

a) √2, 4, 3√2;

b) √3, 3, 2√3;

c) √3, √5, √8.

Answer: a) - yes;

b) - yes;

c) - yes.

**Problem 5**

Find the length of the hypothenuse of a right triangle, if the lengths of the other two sides are:

a)√2 and √3;

b) √5 and √7;

c) √9 and √11.

Answer: a) - √5;

b) - √12;

c) - √20.

**Problem 6**

In the parallelogram ABCD - AB = 33cm, BC = 56cm and AC = 65cm. Check if the parallelogram is rectangle.

Answer: yes.

**Problem 7**

The mid-points of the sides of a rhombus form a rectangle with sides measure $a$ and $b$.
What is the length of the side of the rhombus.

Answer:√a^{2} + b^{2}.

**Problem 8**

The hypothenuse of a right triangle is 10, and one of its other sides is - 8. Find the area of the triangle.

Answer: 24

**Problem 9**

One of the sides of rectangle is 12in, and its diagonal is - 13in. Find the area of the rectangle.

Answer: 60in^{2}.

**Problem 10**

The diagonals of a rhombus are 10 and 4. Find the length of the side of the rhombus.

Answer: $\sqrt{29} \approx 5.4$