What is The Pythagorean Theorem

What Is the Pythagorean Theorem?

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The Pythagorean theorem applies only to right triangles. If \(a\) and \(b\) are legs and \(c\) is the hypotenuse, then \(a^2+b^2=c^2\). The hypotenuse is always the side across from the right angle and is the longest side.

Core Formulas

  • Find a hypotenuse: \(c=\\sqrt{a^2+b^2}\).
  • Find a missing leg: \(a=\\sqrt{c^2-b^2}\).
  • Distance on a coordinate plane: \(d=\\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\).

Worked Example

If the legs are \(6\) and \(8\), then \(c^2=6^2+8^2=36+64=100\), so \(c=10\).

Reference Diagram

The Pythagorean Theorem

The Pythagorean Theorem2

Video Lesson

Original Practice Figures

These saved figures are kept with the lesson for continuity. The exercise text below gives all needed measurements.

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The Pythagorean Theorem

Think of this lesson as more than a rule to memorize. The Pythagorean Theorem is about shape relationships, formulas, and units. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Geometry formulas work because they measure a specific feature: length around, space inside, or space enclosed by a solid. Match the question to the measurement first.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

  • Read what is given and what is being asked.
  • Choose the rule that connects them.
  • Substitute carefully and simplify in small steps.
  • Check the final answer against the original question.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Exercises

Solve each ACT-style practice problem. The questions increase in difficulty.

1) A right triangle has legs 3 and 4. Find the hypotenuse.

2) A right triangle has legs 5 and 12. Find the hypotenuse.

3) A right triangle has hypotenuse 10 and one leg 6. Find the other leg.

4) A right triangle has hypotenuse 17 and one leg 8. Find the other leg.

5) A ladder is 13 ft long and its base is 5 ft from a wall. How high does it reach?

6) A point is 9 units east and 12 units north of another point. What is the distance between the points?

7) A square has side length 10. What is the length of its diagonal?

8) A rectangle has diagonal 25 and width 7. Find its length.

9) A right triangle has legs 9 and 10. Find the hypotenuse exactly.

10) A right triangle has legs 11 and 14. Find the hypotenuse exactly.

11) A ramp rises 8 ft over a horizontal run of 15 ft. How long is the ramp?

12) A right triangle has hypotenuse 29 and one leg 20. Find the other leg.

13) Find the distance between \((-3,4)\) and \((5,19)\).

14) A television screen is 48 in wide and 27 in tall. Find the diagonal length exactly.

15) A right triangle has hypotenuse 41 and one leg 9. Find the other leg.

16) A rectangle has length 30 and diagonal 34. Find its width.

17) A guy wire is attached 24 ft up a pole and anchored 10 ft from the pole. How long is the wire?

18) The legs of a right triangle are \(x\) and \(2x\), and the hypotenuse is \(10\). Find \(x\).

19) A right triangle has legs \(7\\sqrt{2}\) and \(7\\sqrt{2}\). Find the hypotenuse.

20) A diagonal path crosses a rectangular field that is 60 yd by 91 yd. How long is the path?

 
1) Use \(c^2=a^2+b^2\).
\(c^2=3^2+4^2=9+16=25\).
So \(c=\\sqrt{25}=5\) units.
2) Use \(c^2=a^2+b^2\).
\(c^2=5^2+12^2=25+144=169\).
So \(c=\\sqrt{169}=13\) units.
3) Use \(\text{missing leg}^2=c^2-\text{known leg}^2\).
\(x^2=10^2-6^2=100-36=64\).
So \(x=\\sqrt{64}=8\) units.
4) Use \(\text{missing leg}^2=c^2-\text{known leg}^2\).
\(x^2=17^2-8^2=289-64=225\).
So \(x=\\sqrt{225}=15\) units.
5) Use \(\text{missing leg}^2=c^2-\text{known leg}^2\).
\(x^2=13^2-5^2=169-25=144\).
So \(x=\\sqrt{144}=12\) ft.
6) Use \(c^2=a^2+b^2\).
\(c^2=9^2+12^2=81+144=225\).
So \(c=\\sqrt{225}=15\) units.
7) Use \(c^2=a^2+b^2\).
\(c^2=10^2+10^2=100+100=200\).
So \(c=\\sqrt{200}=10\\sqrt{2}\) units.
8) Use \(\text{missing leg}^2=c^2-\text{known leg}^2\).
\(x^2=25^2-7^2=625-49=576\).
So \(x=\\sqrt{576}=24\) units.
9) Use \(c^2=a^2+b^2\).
\(c^2=9^2+10^2=81+100=181\).
So \(c=\\sqrt{181}=\\sqrt{181}\) units.
10) Use \(c^2=a^2+b^2\).
\(c^2=11^2+14^2=121+196=317\).
So \(c=\\sqrt{317}=\\sqrt{317}\) units.
11) Use \(c^2=a^2+b^2\).
\(c^2=8^2+15^2=64+225=289\).
So \(c=\\sqrt{289}=17\) ft.
12) Use \(\text{missing leg}^2=c^2-\text{known leg}^2\).
\(x^2=29^2-20^2=841-400=441\).
So \(x=\\sqrt{441}=21\) units.
13) Use \(c^2=a^2+b^2\).
\(c^2=8^2+15^2=64+225=289\).
So \(c=\\sqrt{289}=17\) units.
14) Use \(c^2=a^2+b^2\).
\(c^2=48^2+27^2=2304+729=3033\).
So \(c=\\sqrt{3033}=3\\sqrt{337}\) in.
15) Use \(\text{missing leg}^2=c^2-\text{known leg}^2\).
\(x^2=41^2-9^2=1681-81=1600\).
So \(x=\\sqrt{1600}=40\) units.
16) Use \(\text{missing leg}^2=c^2-\text{known leg}^2\).
\(x^2=34^2-30^2=1156-900=256\).
So \(x=\\sqrt{256}=16\) units.
17) Use \(c^2=a^2+b^2\).
\(c^2=24^2+10^2=576+100=676\).
So \(c=\\sqrt{676}=26\) ft.
18) Use \(a^2+b^2=c^2\): \(x^2+(2x)^2=10^2\).
Combine like terms: \(5x^2=100\).
Divide by \(5\): \(x^2=20\), so \(x=\\sqrt{20}=2\\sqrt{5}\) units.
19) Use \(c^2=a^2+b^2\).
\(c^2=(7\\sqrt{2})^2+(7\\sqrt{2})^2=98+98=196\).
Thus \(c=\\sqrt{196}=14\) units.
20) Use \(c^2=a^2+b^2\).
\(c^2=60^2+91^2=3600+8281=11881\).
So \(c=\\sqrt{11881}=109\) yd.

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