kuta software - infinite pre-algebra
the pythagorean theorem
do the following lengths form a right triangle?
1) triangle with side lengths 6, 8, 9
2) triangle with side lengths 5, 12, 13
3) right triangle with leg lengths 8, 6, hypotenuse 10
4) triangle with side lengths 3, 4, 6
5) ( a = 6.4 ), ( b = 12 ), ( c = 12.2 )
6) ( a = 2.1 ), ( b = 7.2 ), ( c = 7.5 )
find each missing length to the nearest tenth.
7) right triangle with leg length 4, hypotenuse 8
8) right triangle with leg lengths 6, 3
9) right triangle with leg length 7, hypotenuse 10
10) right triangle with leg lengths 3, 7
11) right triangle with hypotenuse 7, leg length 2
12) right triangle with leg length 2, hypotenuse 6

Question

Accepted Answer

Let's solve problem 1: Check if 6, 8, 9 form a right triangle.

# Explanation:
## Step1: Recall Pythagorean theorem
For a right triangle, $a^2 + b^2 = c^2$ (where $c$ is the longest side). Here, $c = 9$, $a = 6$, $b = 8$.
Calculate $a^2 + b^2$: $6^2 + 8^2 = 36 + 64 = 100$.
Calculate $c^2$: $9^2 = 81$.
## Step2: Compare results
Since $100
eq 81$, the sides do not satisfy the Pythagorean theorem.

# Answer: No

Problem 2: Check 5, 12, 13.

# Explanation:
## Step1: Apply Pythagorean theorem
$c = 13$, $a = 5$, $b = 12$.
$a^2 + b^2 = 5^2 + 12^2 = 25 + 144 = 169$.
$c^2 = 13^2 = 169$.
## Step2: Compare
Since $169 = 169$, they satisfy the theorem.

# Answer: Yes

Problem 3: Check 6, 8, 10 (right - angled triangle as per diagram).

# Explanation:
## Step1: Use Pythagorean theorem
$c = 10$, $a = 6$, $b = 8$.
$a^2 + b^2 = 6^2 + 8^2 = 36+64 = 100$.
$c^2 = 10^2 = 100$.
## Step2: Compare
Since $100 = 100$, it is a right triangle.

# Answer: Yes

Problem 4: Check 3, 4, 6.

# Explanation:
## Step1: Apply Pythagorean theorem
$c = 6$, $a = 3$, $b = 4$.
$a^2 + b^2 = 3^2+4^2 = 9 + 16=25$.
$c^2 = 6^2 = 36$.
## Step2: Compare
Since $25
eq36$, not a right triangle.

# Answer: No

Problem 5: $a = 6.4$, $b = 12$, $c = 12.2$

# Explanation:
## Step1: Calculate $a^2 + b^2$ and $c^2$
$a^2 + b^2=(6.4)^2+12^2 = 40.96+144 = 184.96$.
$c^2=(12.2)^2 = 148.84$.
## Step2: Compare
Since $184.96
eq148.84$, not a right triangle.

# Answer: No

Problem 6: $a = 2.1$, $b = 7.2$, $c = 7.5$

# Explanation:
## Step1: Calculate $a^2 + b^2$ and $c^2$
$a^2 + b^2=(2.1)^2+7.2^2=4.41 + 51.84 = 56.25$.
$c^2=(7.5)^2 = 56.25$.
## Step2: Compare
Since $56.25 = 56.25$, it is a right triangle.

# Answer: Yes

Problem 7: Right triangle with legs 4 and 8, find hypotenuse? Wait, no, the right angle is between 4 and the other leg, hypotenuse is 8? Wait, no, the diagram: right - angled triangle, legs 4 and $x$, hypotenuse 8. Wait, no, let's re - check. If it's a right - angled triangle with one leg 4 and hypotenuse 8, find the other leg.

# Explanation:
## Step1: Let the missing leg be $x$. By Pythagorean theorem, $4^2+x^2 = 8^2$
$x^2=8^2 - 4^2=64 - 16 = 48$
## Step2: Solve for $x$
$x=\sqrt{48}\approx6.9$ (to the nearest tenth)

# Answer: 6.9

Problem 8: Right triangle with legs 6 and 3, find hypotenuse? Wait, no, the right angle is between 3 and the other leg, hypotenuse is 6? Wait, no, the diagram: right - angled triangle, legs 3 and $x$, hypotenuse 6. Wait, no, let's use Pythagorean theorem. If legs are 3 and $x$, hypotenuse 6: $3^2+x^2 = 6^2$→$x^2 = 36 - 9=27$→$x=\sqrt{27}\approx5.2$? Wait, no, maybe the legs are 6 and 3, find the other leg? Wait, the right angle is between 6 and 3? No, the diagram shows right angle, one leg 6, one leg 3, find the hypotenuse.

# Explanation:
## Step1: Let hypotenuse be $c$. $a = 6$, $b = 3$. $c=\sqrt{6^2 + 3^2}=\sqrt{36 + 9}=\sqrt{45}\approx6.7$
Wait, maybe I misread. Let's re - do. If the right - angled triangle has legs 3 and 6, then hypotenuse $c=\sqrt{3^2+6^2}=\sqrt{9 + 36}=\sqrt{45}\approx6.7$

# Answer: 6.7

Problem 9: Right triangle with legs 7 and $x$, hypotenuse 10.

# Explanation:
## Step1: By Pythagorean theorem, $7^2+x^2 = 10^2$
$x^2=100 - 49 = 51$
## Step2: Solve for $x$
$x=\sqrt{51}\approx7.1$

# Answer: 7.1

Problem 10: Right triangle with legs 3 and 7, find hypotenuse.

# Explanation:
## Step1: $c=\sqrt{3^2 + 7^2}=\sqrt{9+49}=\sqrt{58}\approx7.6$

# Answer: 7.6

Problem 11: Right triangle with leg 2 and hypotenuse 7, find the other leg.

# Explanation:
## Step1: Let the missing leg be $x$. $x^2+2^2 = 7^2$
$x^2=49 - 4 = 45$
## Step2: Solve for $x$
$x=\sqrt{45}\approx6.7$

# Answer: 6.7

Problem 12: Right triangle with leg 2 and hypotenuse 6, find the other leg.

# Explanation:
## Step1: Let the missing leg be $x$. $x^2+2^2 = 6^2$
$x^2=36 - 4 = 32$
## Step2: Solve for $x$
$x=\sqrt{32}\approx5.7$

# Answer: 5.7

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Recall Pythagorean theorem

Step2: Compare results

Step1: Apply Pythagorean theorem

Step2: Compare

Step1: Use Pythagorean theorem

Step2: Compare

Answer: