To solve the problem, we need to use the Pythagorean Theorem, which states:
\[
a^2 + b^2 = c^2
\]
where \(a\) and \(b\) are the legs of a right triangle, and \(c\) is the hypotenuse. We will apply this theorem to each labeled triangle in the image to find the missing sides.
Step-by-Step Solution
#### 1.
Triangle with sides 30, 40, and \(d_1\):
This is a right triangle with legs 30 and 40, and hypotenuse \(d_1\).
\[
30^2 + 40^2 = d_1^2
\]
\[
900 + 1600 = d_1^2
\]
\[
2500 = d_1^2
\]
\[
d_1 = \sqrt{2500} = 50
\]
#### 2.
Triangle with sides 20, 21, and \(d_2\):
This is a right triangle with legs 20 and 21, and hypotenuse \(d_2\).
\[
20^2 + 21^2 = d_2^2
\]
\[
400 + 441 = d_2^2
\]
\[
841 = d_2^2
\]
\[
d_2 = \sqrt{841} = 29
\]
#### 3.
Triangle with sides 15, 20, and \(d_3\):
This is a right triangle with legs 15 and 20, and hypotenuse \(d_3\).
\[
15^2 + 20^2 = d_3^2
\]
\[
225 + 400 = d_3^2
\]
\[
625 = d_3^2
\]
\[
d_3 = \sqrt{625} = 25
\]
#### 4.
Triangle with sides 7, 24, and \(d_4\):
This is a right triangle with legs 7 and 24, and hypotenuse \(d_4\).
\[
7^2 + 24^2 = d_4^2
\]
\[
49 + 576 = d_4^2
\]
\[
625 = d_4^2
\]
\[
d_4 = \sqrt{625} = 25
\]
#### 5.
Triangle with sides 12, 16, and \(d_5\):
This is a right triangle with legs 12 and 16, and hypotenuse \(d_5\).
\[
12^2 + 16^2 = d_5^2
\]
\[
144 + 256 = d_5^2
\]
\[
400 = d_5^2
\]
\[
d_5 = \sqrt{400} = 20
\]
#### 6.
Triangle with sides 9, 12, and \(d_6\):
This is a right triangle with legs 9 and 12, and hypotenuse \(d_6\).
\[
9^2 + 12^2 = d_6^2
\]
\[
81 + 144 = d_6^2
\]
\[
225 = d_6^2
\]
\[
d_6 = \sqrt{225} = 15
\]
#### 7.
Triangle with sides 5, 12, and \(d_7\):
This is a right triangle with legs 5 and 12, and hypotenuse \(d_7\).
\[
5^2 + 12^2 = d_7^2
\]
\[
25 + 144 = d_7^2
\]
\[
169 = d_7^2
\]
\[
d_7 = \sqrt{169} = 13
\]
#### 8.
Triangle with sides 8, 15, and \(d_8\):
This is a right triangle with legs 8 and 15, and hypotenuse \(d_8\).
\[
8^2 + 15^2 = d_8^2
\]
\[
64 + 225 = d_8^2
\]
\[
289 = d_8^2
\]
\[
d_8 = \sqrt{289} = 17
\]
#### 9.
Triangle with sides 6, 8, and \(d_9\):
This is a right triangle with legs 6 and 8, and hypotenuse \(d_9\).
\[
6^2 + 8^2 = d_9^2
\]
\[
36 + 64 = d_9^2
\]
\[
100 = d_9^2
\]
\[
d_9 = \sqrt{100} = 10
\]
#### 10.
Triangle with sides 5, 12, and \(d_{10}\):
This is a right triangle with legs 5 and 12, and hypotenuse \(d_{10}\).
\[
5^2 + 12^2 = d_{10}^2
\]
\[
25 + 144 = d_{10}^2
\]
\[
169 = d_{10}^2
\]
\[
d_{10} = \sqrt{169} = 13
\]
Final Answer
\[
\boxed{50, 29, 25, 25, 20, 15, 13, 17, 10, 13}
\]
Parent Tip: Review the logic above to help your child master the concept of pythagorean puzzle worksheet.
