Pythagorean Theorem - Problems

Problem 1

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

Solution. Let the unknown distance be $x.$ Then $x^2 = 40^2 + 28^2 = 1600 + 784 = 2384,$ so ${x = \sqrt{2384} \approx 48.8}$ miles.

Problem 2

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

Solution. $3^2 + 4^2 = 9 + 16 = 25 = 5^2.$ By the converse of the Pythagorean theorem the triangle is right.

Problem 3

The table below gives the distances between the 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: ${9^2 + 12^2 = 225 = 15^2};$
b) no: ${5^2 + 11^2 = 146},$ but ${12^2 = 144};$
c) no: ${8.8^2 + 11.7^2 = 214.33},$ but ${14.8^2 = 219.04};$
d) no: ${5^2 + 8^2 = 89},$ but ${9^2 = 81};$
e) yes: ${9^2 + 40^2 = 1681 = 41^2};$
f) yes: ${4.5^2 + 6^2 = 56.25 = 7.5^2}.$

Problem 4

Check whether a triangle is right if the lengths of its sides are:
a) $\sqrt{2},$ $4,$ $3\sqrt{2};$
b) $\sqrt{3},$ $3,$ $2\sqrt{3};$
c) $\sqrt{3},$ $\sqrt{5},$ $\sqrt{8}.$

Answer: a) yes; b) yes; c) yes.

Problem 5

Find the length of the hypotenuse of a right triangle, if the lengths of the other two sides are:
a) $\sqrt{2}$ and $\sqrt{3};$
b) $\sqrt{5}$ and $\sqrt{7};$
c) $\sqrt{9}$ and $\sqrt{11}.$

Answer: a) $\sqrt{5};$ b) ${\sqrt{12} = 2\sqrt{3}};$ c) ${\sqrt{20} = 2\sqrt{5}}.$

Problem 6

In the parallelogram $ABCD$ we have ${AB = 33}$ cm, ${BC = 56}$ cm and ${AC = 65}$ cm. Check if the parallelogram is a rectangle.

Answer: yes.

Problem 7

The midpoints of the sides of a rhombus form a rectangle with sides that measure $a$ and $b.$ What is the length of the side of the rhombus?

Answer: $\sqrt{a^2 + b^2}.$

Problem 8

The hypotenuse 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 a rectangle is 12 in, and its diagonal is 13 in. Find the area of the rectangle.

Answer: $60\ \text{in}^2.$

Problem 10

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

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