Lesson 6
Finding Side Lengths of Triangles
Problem 1
Here is a diagram of an acute triangle and three squares.
![An acute triangle with squares along each side of the triangle.](https://cms-im.s3.amazonaws.com/FPwQj4gdi9PW7DJqvNDLLp1F?response-content-disposition=inline%3B%20filename%3D%228-8.8.B.PP.Image.05.png%22%3B%20filename%2A%3DUTF-8%27%278-8.8.B.PP.Image.05.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T172215Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=cb7627abfd0ce923115f8973be282f7e252cb552f872f55fba5d329861b70ba9)
Priya says the area of the large unmarked square is 26 square units because \(9+17=26\). Do you agree? Explain your reasoning.
Solution
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Problem 2
\(m\), \(p\), and \(z\) represent the lengths of the three sides of this right triangle.
![Right triangle, legs = p, z, hypotenuse = m](https://cms-im.s3.amazonaws.com/vqJ9KTX61yk7Q6TmFAmMuPHf?response-content-disposition=inline%3B%20filename%3D%228-8.8.B6.PP.Image.0003.png%22%3B%20filename%2A%3DUTF-8%27%278-8.8.B6.PP.Image.0003.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T172215Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=0c84c0ae23bf0517dec8f55668767f6a6264118ee6d7ad4bf91f3f9f3c4ce378)
Select all the equations that represent the relationship between \(m\), \(p\), and \(z\).
\(m^2+p^2=z^2\)
\(m^2=p^2+z^2\)
\(m^2=z^2+p^2\)
\(p^2+m^2=z^2\)
\(z^2+p^2=m^2\)
\(p^2+z^2=m^2\)
Solution
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Problem 3
The lengths of the three sides are given for several right triangles. For each, write an equation that expresses the relationship between the lengths of the three sides.
- 10, 6, 8
- \(\sqrt5, \sqrt3, \sqrt8\)
- 5, \(\sqrt5, \sqrt{30}\)
- 1, \(\sqrt{37}\), 6
- 3, \(\sqrt{2}, \sqrt7\)
Solution
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Problem 4
Order the following expressions from least to greatest.
\(25\div 10\)
\(250,\!000 \div 1,\!000\)
\(2.5 \div 1,\!000\)
\(0.025\div 1\)
Solution
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(From Unit 4, Lesson 1.)Problem 5
Which is the best explanation for why \(\text-\sqrt{10}\) is irrational?
\(\text- \sqrt{10}\) is irrational because it is not rational.
\(\text- \sqrt{10}\) is irrational because it is less than zero.
\(\text- \sqrt{10}\) is irrational because it is not a whole number.
\(\text- \sqrt{10}\) is irrational because if I put \(\text- \sqrt{10}\) into a calculator, I get -3.16227766, which does not make a repeating pattern.
Solution
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(From Unit 8, Lesson 3.)Problem 6
A teacher tells her students she is just over 1 and \(\frac{1}{2}\) billion seconds old.
- Write her age in seconds using scientific notation.
- What is a more reasonable unit of measurement for this situation?
- How old is she when you use a more reasonable unit of measurement?
Solution
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(From Unit 7, Lesson 15.)