Converse of Pythagorean Theorem Worksheet - Math Monks - Free Printable

To determine if a triangle is a right triangle, we use the Converse of the Pythagorean Theorem. This rule states that if the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides ($a^2 + b^2 = c^2$), then it is a right triangle.

Here is the step-by-step check for each problem:

Part 1: Are the triangles below right triangles?

1) Sides: 9, 12, 15
* Longest side ($c$): 15
* Other sides ($a, b$): 9, 12
* Check: $9^2 + 12^2 = 81 + 144 = 225$
* Hypotenuse squared: $15^2 = 225$
* Since $225 = 225$, this is a right triangle.
* Answer: Yes (Right angle is between sides 9 and 12).

2) Sides: 18, 24, 30
* Longest side ($c$): 30
* Other sides ($a, b$): 18, 24
* Check: $18^2 + 24^2 = 324 + 576 = 900$
* Hypotenuse squared: $30^2 = 900$
* Since $900 = 900$, this is a right triangle.
* Answer: Yes (Right angle is between sides 18 and 24).

3) Sides: 7, 13, 17
* Longest side ($c$): 17
* Other sides ($a, b$): 7, 13
* Check: $7^2 + 13^2 = 49 + 169 = 218$
* Hypotenuse squared: $17^2 = 289$
* Since $218 \neq 289$, this is NOT a right triangle.
* Answer: No

4) Sides: 8, 7, $\sqrt{113}$
* Longest side ($c$): $\sqrt{113}$ (because $\sqrt{113} \approx 10.6$, which is larger than 8 and 7)
* Other sides ($a, b$): 8, 7
* Check: $8^2 + 7^2 = 64 + 49 = 113$
* Hypotenuse squared: $(\sqrt{113})^2 = 113$
* Since $113 = 113$, this is a right triangle.
* Answer: Yes (Right angle is between sides 8 and 7).

5) Sides: 3.9, 10.5, 11
* Longest side ($c$): 11
* Other sides ($a, b$): 3.9, 10.5
* Check: $3.9^2 + 10.5^2 = 15.21 + 110.25 = 125.46$
* Hypotenuse squared: $11^2 = 121$
* Since $125.46 \neq 121$, this is NOT a right triangle.
* Answer: No

6) Sides: 4, 8, 8.9
* Longest side ($c$): 8.9
* Other sides ($a, b$): 4, 8
* Check: $4^2 + 8^2 = 16 + 64 = 80$
* Hypotenuse squared: $8.9^2 = 79.21$
* Since $80 \neq 79.21$, this is NOT a right triangle.
* Answer: No

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Part 2: Do the following 3 sides make a right triangle?

7) Sides: $10, 15, 5\sqrt{13}$
* First, estimate the values to find the longest side. $\sqrt{13} \approx 3.6$, so $5\sqrt{13} \approx 18$. This is the longest side ($c$).
* Check: $10^2 + 15^2 = 100 + 225 = 325$
* Hypotenuse squared: $(5\sqrt{13})^2 = 5^2 \cdot (\sqrt{13})^2 = 25 \cdot 13 = 325$
* Since $325 = 325$, it is a right triangle.
* Answer: Yes

8) Sides: $6, 2, 2\sqrt{10}$
* Estimate: $\sqrt{10} \approx 3.16$, so $2\sqrt{10} \approx 6.32$. This is the longest side ($c$).
* Check: $6^2 + 2^2 = 36 + 4 = 40$
* Hypotenuse squared: $(2\sqrt{10})^2 = 2^2 \cdot 10 = 4 \cdot 10 = 40$
* Since $40 = 40$, it is a right triangle.
* Answer: Yes

9) Sides: $2\sqrt{14}, 13, 15$
* Estimate: $\sqrt{14} \approx 3.74$, so $2\sqrt{14} \approx 7.48$. The sides are approx 7.48, 13, 15. Longest is 15 ($c$).
* Check: $(2\sqrt{14})^2 + 13^2 = (4 \cdot 14) + 169 = 56 + 169 = 225$
* Hypotenuse squared: $15^2 = 225$
* Since $225 = 225$, it is a right triangle.
* Answer: Yes

10) Sides: $5, 10, 14$
* Longest side ($c$): 14
* Check: $5^2 + 10^2 = 25 + 100 = 125$
* Hypotenuse squared: $14^2 = 196$
* Since $125 \neq 196$, it is NOT a right triangle.
* Answer: No

11) Sides: $6, 5, \sqrt{61}$
* Estimate: $\sqrt{61}$ is between $\sqrt{49}=7$ and $\sqrt{64}=8$ (approx 7.8). So $\sqrt{61}$ is the longest side ($c$).
* Check: $6^2 + 5^2 = 36 + 25 = 61$
* Hypotenuse squared: $(\sqrt{61})^2 = 61$
* Since $61 = 61$, it is a right triangle.
* Answer: Yes

12) Sides: $\sqrt{3}, 2, 3$
* Estimate: $\sqrt{3} \approx 1.73$. Sides are approx 1.73, 2, 3. Longest is 3 ($c$).
* Check: $(\sqrt{3})^2 + 2^2 = 3 + 4 = 7$
* Hypotenuse squared: $3^2 = 9$
* Since $7 \neq 9$, it is NOT a right triangle.
* Answer: No

Final Answer:
Part 1:
1) Yes
2) Yes
3) No
4) Yes
5) No
6) No

Part 2:
7) Yes
8) Yes
9) Yes
10) No
11) Yes
12) No