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 x2 = 402 + 282 = 1600 + 784 = 2384
x2 = 2384
$x = \sqrt{2384} \approx 49$ miles.

Problem 2
Prove that the triangle is right if its sides measure 3in, 4in and 5in.

Solution 32 + 42 = 9 + 16 = 25 = 52. 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:√a2 + b2.

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: 60in2.

Problem 10
The diagonals of a rhombus are 10 and 4. Find the length of the side of the rhombus.
Answer: $\sqrt{29} \approx 5.4$

Other resources

Other problems involving the Pythagorean theorem

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