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Mathematics LibreTexts

4.4: Pythagorean Theorem

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In a right triangle, the sides of the right angle are called the legs of the triangle and the remaining side is called the hypotenuse. In Figure 4.4.1, side AC and BC are the legs and side AB is the hypotenuse.

clipboard_e90204b4e2ed8e21086d6caac07a505e7.png
Figure 4.4.1: A right triangle.

The following is one of the most famous theorems in mathematics.

Theorem 4.4.1: Pythagorean Theorem

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. That is,

leg2+leg2=hypotenuse2

Thus, for the sides of the triangle in Figure 4.4.1,

a2+b2=c2

Before we prove Theorem 4.4.1, we will give several examples.

Example 4.4.1

Find x

clipboard_e17d5483ff803fa5185b945be12dc7750.png

Solution

leg2+leg2=hyp232+42=x29+16=x225=x25=x

Check:

屏幕快照 2020-11-17 下午7.38.11.png

Answer: x=5.

Example 4.4.2

Find x:

clipboard_eb7ac06257600a605c9b3a04df0b7cc18.png

Solution

leg2+leg2=hyp252+x2=10225+x2=100x2=75x=75=253=53

Check:

屏幕快照 2020-11-17 下午7.41.41.png

Answer: x=53.

Example 4.4.3

Find x:

clipboard_e14936d6e84046a616dccef57301f504c.png

Solution

leg2+leg2=hyp252+52=x225+25=x250=x2x=50=252=52

Check:

屏幕快照 2020-11-17 下午7.43.37.png

Answer: x=52.

Example 4.4.4

Find x

clipboard_eaaa2e5f9f9b43e63a9b2017ad35dcbbc.png

Solution

leg2+leg2=hyp2x2+(x+1)2=(x+2)2x2+x2+2x+1=x2+4x+4x2+x2+2x+1x24x4=0x22x3=0(x3)(x+1)=0

x3=0x=3 x+1=0x=1

We reject x=1 because AC=x cannot be negative.

Check, x=3:

屏幕快照 2020-11-17 下午7.49.44.png

Answer: x=3.

We will now restate and prove Theorem 4.4.1:

Theorem 4.4.1 Pythagorean Theorem

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. That is,

leg2+leg2=hypotenuse2.

In Figure 4.4.1,

a2+b2=c2.

clipboard_e23b400bdbb092d0a60d8eaa87f876363.png
Figure 4.4.1. A right triangle.
clipboard_e8252c1b03aac055dcf4fd9ba249b6087.png
Figure 4.4.2. Draw CD perpendicular to AB.
Proof

In Figure 4.4.1, draw CD perpendicular to AB. Let x=AD. Then BD=cx (Figure 4.4.2). As in Example 4.4.3, section 4.2, ABCACD and ABCCBD. From these two similarities we obtain two proportions:

屏幕快照 2020-11-17 下午7.57.30.png

The converse of the Pythagorean Theorem also holds:

Theorem 4.4.2 (converse of the Pythagorean Theorem).

In a triangle, if the square of one side is equal to the sun of the squares of the other two sides then the triangle is a right triangle.

In Figure 4.4.3, if c2=a2+b2 then ABC is a right triangle with C=90.

clipboard_e32efa960b617195103c9bc7f5ef0214e.png
Figure 4.4.3: If c2=a2+b2 then C=90.
Proof

Draw a new triangle, DEF, so that F=90, d=a, and e=b (Figure 4.4.4). DEF is a right triangle, so by Theorem 4.4.1, f2=d2+e2. We have f2=d2+e2=a2+b2=c2 and therefore f=c. Therefore ABCDEF because SSS=SSS. Therefore, C+F=90.

屏幕快照 2020-11-17 下午8.06.53.png
Figure 4.4.4: Given ABC, draw DEF so that F=90, d=a and e=b.
Example 4.4.5

Is ABC a right triangle?

屏幕快照 2020-11-17 下午8.09.01.png

Solution

AC2=72=49

BC2=92=81

AB2=(130)2=130

49+81=130.

so by Theorem 4.4.2, ABC is a right triangle.

Answer: yes.

Example 4.4.6

Find x and AB:

屏幕快照 2020-11-17 下午8.12.34.png

Solution

x2+122=132x2+144=169x2=169144x2=25x=5

CDEF is a rectangle so EF=CD=20 and CF=DE=12. Therefore FB=5 and AB=AE+EF+FB=5+20+5=30.

Answer: x=5, AB=30.

Example 4.4.7

Find x, AC and BD:

屏幕快照 2020-11-17 下午8.18.09.png

Solution

ABCD is a rhombus. The diagonals of a rhombus are perpendicular and bisect each other.

62+82=x236+64=x2100=x210=x

AC=8+8=16,BD=6+6=12.

Answer: x=10,AC=16,BD=12.

Example 4.4.8

A ladder 39 feet long leans against a building, How far up the side of the building does the ladder reach if the foot of the ladder is 15 feet from the building?

屏幕快照 2020-11-17 下午8.21.43.png

Solution

leg2+leg2=hyp2x2+152=392x2+225=1521x2=1521225x2=1296x=1296=36

Answer: 36 feet.

Historical Note

Pythagoras (c. 582 - 507 B.C.) was not the first to discover the theorem which bears his name. It was known long before his time by the Chinese, the Babylonians, and perhaps also the Egyptians and the Hindus, According to tradition, Pythagoras was the first to give a nroof of the theorem, His proof probably made use of areas, like the one suggested. In Figure 4.4.5 below, (each square contains four congruent right triangles with sides of lengths a, b, and c, In addition the square on the left contains a square with side a and a square with side b while the one on the right contains a square with side c.)

屏幕快照 2020-11-17 下午8.27.10.png
Figure 4.4.5: Pythagoras may have proved a2+b2=c2 in this way.
Since the time of Pythagoras, at least several hundred different proofs of the Pythagorean Theorem have been proposed, Pythagoras was the founder of the Pythagorean school, a secret religious society devoted to the study of philosophy, mathematics, and science. Its membership was a select group, which tended to keep the discoveries and practices of the society secret from outsiders. The Pythagoreans believed that numbers were the ultimate components of the universe and that all physical relationships could be expressed with whole numbers, This belief was prompted in part by their discovery that the notes of the musical scale were related by numerical ratios. The Pythagoreans made important contributions to medicine, physics, and astronomy, In geometry, they are credited with the angle s

um theorem for triangles, the properties of parallel lines, and the theory of similar triangles and proportions.

Problems

1 - 10. Find x. Leave answers in simplest radical form.

1.

Screen Shot 2020-11-17 at 8.44.19 PM.png

2.

Screen Shot 2020-11-17 at 8.44.39 PM.png

3.

Screen Shot 2020-11-17 at 8.45.25 PM.png

4.

Screen Shot 2020-11-17 at 8.45.43 PM.png

5.

Screen Shot 2020-11-17 at 8.46.01 PM.png

6.

Screen Shot 2020-11-17 at 8.46.20 PM.png

7.

Screen Shot 2020-11-17 at 8.46.36 PM.png

8.

Screen Shot 2020-11-17 at 8.46.50 PM.png

9.

Screen Shot 2020-11-17 at 8.47.05 PM.png

10.

Screen Shot 2020-11-17 at 8.47.21 PM.png

11 - 14. Find x and all sides of the triangle:

11.

Screen Shot 2020-11-17 at 8.47.45 PM.png

12.

Screen Shot 2020-11-17 at 8.48.03 PM.png

13.

Screen Shot 2020-11-17 at 8.48.17 PM.png

14.

Screen Shot 2020-11-17 at 8.48.44 PM.png

15 - 16. Find x:

15.

Screen Shot 2020-11-17 at 8.49.03 PM.png

16.

Screen Shot 2020-11-17 at 8.49.18 PM.png

17. Find x and AB.

Screen Shot 2020-11-17 at 8.49.42 PM.png

18. Find x:

Screen Shot 2020-11-17 at 8.49.59 PM.png

19. Find x,AC and BD:

Screen Shot 2020-11-17 at 8.50.15 PM.png

20. Find x,AC and BD:

Screen Shot 2020-11-17 at 8.50.56 PM.png

21. Find x and y:

Screen Shot 2020-11-17 at 8.51.15 PM.png

22. Find x, AC and BD:

Screen Shot 2020-11-17 at 8.51.34 PM.png

23. Find x,AB and BD:

Screen Shot 2020-11-17 at 8.51.56 PM.png

24. Find x,AB and AD:

Screen Shot 2020-11-17 at 8.52.15 PM.png

25 - 30. Is ABC a right triangle?

25.

Screen Shot 2020-11-17 at 8.52.34 PM.png

26.

Screen Shot 2020-11-17 at 8.52.49 PM.png

27.

Screen Shot 2020-11-17 at 8.53.26 PM.png

28.

Screen Shot 2020-11-17 at 8.53.42 PM.png

29.

Screen Shot 2020-11-17 at 8.53.57 PM.png

30.

Screen Shot 2020-11-17 at 8.54.13 PM.png

31. A ladder 25 feet long leans against a building, How far up the side of the building does the ladder reach if the foot of the ladder is 7 feet from the building?

32. A man travels 24 miles east and then 10 miles north. At the end of his journey how far is he from his starting point?

33. Can a table 9 feet wide (with its legs folded) fit through a rectangular doorway 4 feet by 8 feet?

Screen Shot 2020-11-17 at 8.54.30 PM.png

34. A baseball diamond is a square 90 feet on each side, Find the distance from home plate to second base (leave answer in simplest radical form).

Screen Shot 2020-11-17 at 8.55.11 PM.png


This page titled 4.4: Pythagorean Theorem is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Henry Africk (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform.

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