To solve this problem, we need to find the value of x for each triangle using the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as:
\[ c^2 = a^2 + b^2 \]
where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.
Let's solve for each triangle:
1.
Triangle 1:
- Sides: 5, 12, x
- Hypotenuse: x
- Equation: \( x^2 = 5^2 + 12^2 \)
- Calculation: \( x^2 = 25 + 144 = 169 \)
- \( x = \sqrt{169} = 13 \)
2.
Triangle 2:
- Sides: 8, 15, x
- Hypotenuse: x
- Equation: \( x^2 = 8^2 + 15^2 \)
- Calculation: \( x^2 = 64 + 225 = 289 \)
- \( x = \sqrt{289} = 17 \)
3.
Triangle 3:
- Sides: 9, 12, x
- Hypotenuse: x
- Equation: \( x^2 = 9^2 + 12^2 \)
- Calculation: \( x^2 = 81 + 144 = 225 \)
- \( x = \sqrt{225} = 15 \)
4.
Triangle 4:
- Sides: 10, 24, x
- Hypotenuse: x
- Equation: \( x^2 = 10^2 + 24^2 \)
- Calculation: \( x^2 = 100 + 576 = 676 \)
- \( x = \sqrt{676} = 26 \)
5.
Triangle 5:
- Sides: 12, 16, x
- Hypotenuse: x
- Equation: \( x^2 = 12^2 + 16^2 \)
- Calculation: \( x^2 = 144 + 256 = 400 \)
- \( x = \sqrt{400} = 20 \)
6.
Triangle 6:
- Sides: 15, 20, x
- Hypotenuse: x
- Equation: \( x^2 = 15^2 + 20^2 \)
- Calculation: \( x^2 = 225 + 400 = 625 \)
- \( x = \sqrt{625} = 25 \)
7.
Triangle 7:
- Sides: 18, 24, x
- Hypotenuse: x
- Equation: \( x^2 = 18^2 + 24^2 \)
- Calculation: \( x^2 = 324 + 576 = 900 \)
- \( x = \sqrt{900} = 30 \)
8.
Triangle 8:
- Sides: 20, 21, x
- Hypotenuse: x
- Equation: \( x^2 = 20^2 + 21^2 \)
- Calculation: \( x^2 = 400 + 441 = 841 \)
- \( x = \sqrt{841} = 29 \)
9.
Triangle 9:
- Sides: 24, 32, x
- Hypotenuse: x
- Equation: \( x^2 = 24^2 + 32^2 \)
- Calculation: \( x^2 = 576 + 1024 = 1600 \)
- \( x = \sqrt{1600} = 40 \)
10.
Triangle 10:
- Sides: 30, 40, x
- Hypotenuse: x
- Equation: \( x^2 = 30^2 + 40^2 \)
- Calculation: \( x^2 = 900 + 1600 = 2500 \)
- \( x = \sqrt{2500} = 50 \)
11.
Triangle 11:
- Sides: 36, 48, x
- Hypotenuse: x
- Equation: \( x^2 = 36^2 + 48^2 \)
- Calculation: \( x^2 = 1296 + 2304 = 3600 \)
- \( x = \sqrt{3600} = 60 \)
12.
Triangle 12:
- Sides: 40, 42, x
- Hypotenuse: x
- Equation: \( x^2 = 40^2 + 42^2 \)
- Calculation: \( x^2 = 1600 + 1764 = 3364 \)
- \( x = \sqrt{3364} = 58 \)
Now, we match these values with the options given:
- Triangle 1: 13 (Option D)
- Triangle 2: 17 (Option A)
- Triangle 3: 15 (Option B)
- Triangle 4: 26 (Option C)
- Triangle 5: 20 (Option E)
- Triangle 6: 25 (Option F)
- Triangle 7: 30 (Option G)
- Triangle 8: 29 (Option H)
- Triangle 9: 40 (Option I)
- Triangle 10: 50 (Option J)
- Triangle 11: 60 (Option K)
- Triangle 12: 58 (Option L)
Final Answer:
D, A, B, C, E, F, G, H, I, J, K, L
Parent Tip: Review the logic above to help your child master the concept of triangle sum theorem worksheet answers.
![Printable Triangle Sum Theorem Worksheets [PDFs] Brighterly](/images2/triangle-sum-theorem-worksheet-answers/triangle-sum-theorem-worksheet-answers-11.jpg)
