{"id":3054,"date":"2023-05-19T16:39:27","date_gmt":"2023-05-19T16:39:27","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=3054"},"modified":"2025-08-26T03:26:11","modified_gmt":"2025-08-26T03:26:11","slug":"triangles-learn-it-4","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/triangles-learn-it-4\/","title":{"raw":"Triangles: Learn It 4","rendered":"Triangles: Learn It 4"},"content":{"raw":"

Using the Pythagorean Theorem<\/h2>\r\n

The Pythagorean Theorem <\/strong>is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around [latex]500[\/latex] BCE.<\/p>\r\n

Remember that a right triangle has a [latex]90^\\circ [\/latex] angle, which we usually mark with a small square in the corner. The side of the triangle opposite the [latex]90^\\circ [\/latex] angle is called the hypotenuse<\/strong>, and the other two sides are called the legs.<\/p>\r\n

\r\n[caption id=\"\" align=\"aligncenter\" width=\"561\"]\"Three Figure 1. These right triangles have two legs and a hypotenuse[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.<\/p>\r\n

\r\n
\r\n

the Pythagorean Theorem<\/h3>\r\n

In any right triangle [latex]\\Delta ABC[\/latex],<\/p>\r\n

[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]<\/p>\r\n

 <\/p>\r\n

where [latex]c[\/latex] is the length of the hypotenuse [latex]a[\/latex] and [latex]b[\/latex] are the lengths of the legs.<\/p>\r\n

 <\/p>\r\n

\"A<\/center>\r\n

 <\/p>\r\n<\/div>\r\n<\/section>\r\n

\r\n
\r\n

To solve problems that use the Pythagorean Theorem, we will need to find square roots. Recall the notation [latex]\\sqrt{m}[\/latex] and that it is defined in this way:<\/p>\r\n

[latex]\\text{If }m={n}^{2},\\text{ then }\\sqrt{m}=n\\text{ for }n\\ge 0[\/latex]<\/p>\r\n

For example, [latex]\\sqrt{25}[\/latex] is [latex]5[\/latex] because [latex]{5}^{2}=25[\/latex].<\/p>\r\n

We will use this definition of square roots to solve for the length of a side in a right triangle.<\/p>\r\n<\/div>\r\n<\/section>\r\n

Use the Pythagorean Theorem to find the length of the hypotenuse.
\r\n
\"Right<\/center>
\r\n[reveal-answer q=\"57881\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"57881\"]\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Step 1. Read<\/strong> the problem.<\/td>\r\n <\/td>\r\n<\/tr>\r\n
Step 2. Identify<\/strong> what you are looking for.<\/td>\r\nthe length of the hypotenuse of the triangle<\/td>\r\n<\/tr>\r\n
Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\r\n\r\n

Let [latex]c=\\text{the length of the hypotenuse}[\/latex]<\/p>\r\n

\"Right<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

Step 4. Translate.<\/strong><\/p>\r\n

Write the appropriate formula. Substitute.<\/p>\r\n<\/td>\r\n

\r\n

[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]<\/p>\r\n

[latex]{3}^{2}+{4}^{2}={c}^{2}[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

Step 5. Solve<\/strong> the equation.<\/td>\r\n\r\n

[latex]9+16={c}^{2}[\/latex]<\/p>\r\n

[latex]25={c}^{2}[\/latex]<\/p>\r\n

[latex]\\sqrt{25}={c}^{2}[\/latex]<\/p>\r\n

[latex]5=c[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

Step 6. Check.<\/strong><\/p>\r\n<\/td>\r\n

\r\n

[latex]{3}^{2}+{4}^{2}={\\color{red}{5}}^{2}[\/latex]<\/p>\r\n

[latex]9+16\\stackrel{?}{=}25[\/latex]<\/p>\r\n

[latex]25+25\\checkmark[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

Step 7. Answer<\/strong> the question.<\/td>\r\nThe length of the hypotenuse is [latex]5[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

[\/hidden-answer]<\/p>\r\n<\/section>\r\n

[ohm2_question hide_question_numbers=1]7037[\/ohm2_question]<\/section>\r\n
Use the Pythagorean Theorem to find the length of the longer leg.
\r\n
\"A<\/center>\r\n

 <\/p>\r\n
\r\n[reveal-answer q=\"477507\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"477507\"]\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Step 1. Read<\/strong> the problem.<\/td>\r\n <\/td>\r\n<\/tr>\r\n
Step 2. Identify<\/strong> what you are looking for.<\/td>\r\nThe length of the leg of the triangle<\/td>\r\n<\/tr>\r\n
Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\r\n\r\n

Let [latex]b=\\text{the leg of the triangle}[\/latex]<\/p>\r\n

Label side b<\/em><\/p>\r\n

\"A<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

Step 4. Translate.<\/strong><\/p>\r\n

Write the appropriate formula. Substitute.<\/p>\r\n<\/td>\r\n

\r\n

[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]<\/p>\r\n

[latex]{5}^{2}+{b}^{2}={13}^{2}[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

Step 5. Solve<\/strong> the equation. Isolate the variable term. Use the definition of the square root.<\/p>\r\n

Simplify.<\/p>\r\n<\/td>\r\n

\r\n

[latex]25+{b}^{2}=169[\/latex]<\/p>\r\n

[latex]{b}^{2}=144[\/latex]<\/p>\r\n

[latex]{b}^{2}=\\sqrt{144}[\/latex]<\/p>\r\n

[latex]b=12[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

Step 6. Check.<\/strong><\/p>\r\n<\/td>\r\n

\r\n

[latex]{5}^{2}+{\\color{red}{12}}^{2}\\stackrel{?}{=}{13}^{2}[\/latex]<\/p>\r\n

[latex]25+144\\stackrel{?}{=}169[\/latex]<\/p>\r\n

[latex]169=169\\checkmark[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

Step 7. Answer<\/strong> the question.<\/td>\r\nThe length of the leg is [latex]12[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

[\/hidden-answer]<\/p>\r\n<\/section>\r\n

We can use the Pythagorean Theorem to solve real-world application problems. Let's try a few examples.<\/p>\r\n

Kelvin is building a gazebo and wants to brace each corner by placing a [latex]\\text{10-inch}[\/latex] wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.
\"A<\/center>\r\n

 <\/p>\r\n

[reveal-answer q=\"730318\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"730318\"]<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Step 1. Read<\/strong> the problem.<\/td>\r\n\u00a0<\/td>\r\n<\/tr>\r\n
Step 2. Identify<\/strong> what you are looking for.<\/td>\r\nthe distance from the corner that the bracket should be attached<\/td>\r\n<\/tr>\r\n
Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\r\n\r\n

Let x<\/em> = the distance from the corner<\/p>\r\n

\"A<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

Step 4. Translate.<\/strong><\/p>\r\n

Write the appropriate formula. Substitute.<\/p>\r\n<\/td>\r\n

\r\n

[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]<\/p>\r\n

[latex]{x}^{2}+{x}^{2}={10}^{2}[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

Step 5. Solve<\/strong> the equation.<\/p>\r\n

Isolate the variable.<\/p>\r\n

Use the definition of the square root.<\/p>\r\n

Simplify. Approximate to the nearest tenth.<\/p>\r\n<\/td>\r\n

\r\n

[latex]2x^2=100[\/latex]<\/p>\r\n

[latex]x^2=50[\/latex]<\/p>\r\n

[latex]x=\\sqrt{50}[\/latex]<\/p>\r\n

[latex]x\\approx{7.1}[\/latex]<\/p>\r\n

 <\/p>\r\n<\/td>\r\n<\/tr>\r\n

\r\n

Step 6. Check.<\/strong><\/p>\r\n<\/td>\r\n

\r\n

[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]<\/p>\r\n

[latex]({\\color{red}{7.1}})^2+({\\color{red}{7.1}})^{2}\\stackrel{\\text{?}}{\\approx}{10}^{2}[\/latex]<\/p>\r\n

[latex]50.41+50.41=100.82\\approx{100}\\quad\\checkmark[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n

Step 7. Answer<\/strong> the question.<\/td>\r\nKelvin should fasten each piece of wood approximately [latex]7.1[\/latex]\" from the corner.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

[\/hidden-answer]<\/p>\r\n<\/section>\r\n

[ohm2_question hide_question_numbers=1]7040[\/ohm2_question]<\/section>\r\n
[ohm2_question hide_question_numbers=1]7041[\/ohm2_question]<\/section>","rendered":"

Using the Pythagorean Theorem<\/h2>\n

The Pythagorean Theorem <\/strong>is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around [latex]500[\/latex] BCE.<\/p>\n

Remember that a right triangle has a [latex]90^\\circ[\/latex] angle, which we usually mark with a small square in the corner. The side of the triangle opposite the [latex]90^\\circ[\/latex] angle is called the hypotenuse<\/strong>, and the other two sides are called the legs.<\/p>\n

\n
\"Three
Figure 1. These right triangles have two legs and a hypotenuse<\/figcaption><\/figure>\n<\/div>\n

 <\/p>\n

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.<\/p>\n

\n
\n

the Pythagorean Theorem<\/h3>\n

In any right triangle [latex]\\Delta ABC[\/latex],<\/p>\n

[latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]<\/p>\n

 <\/p>\n

where [latex]c[\/latex] is the length of the hypotenuse [latex]a[\/latex] and [latex]b[\/latex] are the lengths of the legs.<\/p>\n

 <\/p>\n

\"A<\/div>\n

 <\/p>\n<\/div>\n<\/section>\n

\n
\n

To solve problems that use the Pythagorean Theorem, we will need to find square roots. Recall the notation [latex]\\sqrt{m}[\/latex] and that it is defined in this way:<\/p>\n

[latex]\\text{If }m={n}^{2},\\text{ then }\\sqrt{m}=n\\text{ for }n\\ge 0[\/latex]<\/p>\n

For example, [latex]\\sqrt{25}[\/latex] is [latex]5[\/latex] because [latex]{5}^{2}=25[\/latex].<\/p>\n

We will use this definition of square roots to solve for the length of a side in a right triangle.<\/p>\n<\/div>\n<\/section>\n

Use the Pythagorean Theorem to find the length of the hypotenuse.<\/p>\n
\"Right<\/div>\n
<\/div>\n