Solved on Jan 23, 2024

Determine which of the following side lengths cannot represent a right triangle: $6,8,10$ ; $7,24,25$ ; $10,24,26$ ; $15,25,35$ .

STEP 1

Assumptions
1. A set of three numbers represents the lengths of the sides of a triangle.
2. To determine if the numbers can represent a right triangle, we will use the Pythagorean theorem which states that in a right 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.
3. The Pythagorean theorem is represented by the equation $a^2 + b^2 = c^2$ , where $c$ is the length of the hypotenuse and $a$ and $b$ are the lengths of the other two sides.
4. We will check each set of numbers to see if they satisfy the Pythagorean theorem.

STEP 2

Check option A: $6, 8, 10$ .
We will assume the longest side is the hypotenuse and apply the Pythagorean theorem.
$a^2 + b^2 = c^2$
$6^2 + 8^2 = 10^2$

STEP 3

Calculate the squares of the sides for option A.
$36 + 64 = 100$

STEP 4

Check if the sum of the squares of the two shorter sides equals the square of the longest side for option A.
$36 + 64 = 100$
$100 = 100$
Since the equation holds true, option A can represent a right triangle.

STEP 5

Check option B: $7, 24, 25$ .
Again, we will assume the longest side is the hypotenuse and apply the Pythagorean theorem.
$a^2 + b^2 = c^2$
$7^2 + 24^2 = 25^2$

STEP 6

Calculate the squares of the sides for option B.
$49 + 576 = 625$

STEP 7

Check if the sum of the squares of the two shorter sides equals the square of the longest side for option B.
$49 + 576 = 625$
$625 = 625$
Since the equation holds true, option B can represent a right triangle.

STEP 8

Check option C: $10, 24, 26$ .
Again, we will assume the longest side is the hypotenuse and apply the Pythagorean theorem.
$a^2 + b^2 = c^2$
$10^2 + 24^2 = 26^2$

STEP 9

Calculate the squares of the sides for option C.
$100 + 576 = 676$

STEP 10

Check if the sum of the squares of the two shorter sides equals the square of the longest side for option C.
$100 + 576 = 676$
$676 \neq 676$
Since the equation does not hold true (the sum of the squares of the two shorter sides does not equal the square of the longest side), option C cannot represent a right triangle.

STEP 11

Check option D: $15, 25, 35$ .
Again, we will assume the longest side is the hypotenuse and apply the Pythagorean theorem.
$a^2 + b^2 = c^2$
$15^2 + 25^2 = 35^2$

STEP 12

Calculate the squares of the sides for option D.
$225 + 625 = 1225$

STEP 13

Check if the sum of the squares of the two shorter sides equals the square of the longest side for option D.
$225 + 625 = 1225$
$850 \neq 1225$
Since the equation does not hold true (the sum of the squares of the two shorter sides does not equal the square of the longest side), option D cannot represent a right triangle.

STEP 14

By comparing the results from STEP_4, STEP_7, STEP_10, and STEP_13, we can conclude that options A and B satisfy the Pythagorean theorem and can represent right triangles, while options C and D do not satisfy the Pythagorean theorem.
Therefore, the numbers that cannot represent a right triangle are from options C and D. However, since we need to choose one option that cannot represent a right triangle, the answer is option C: $10, 24, 26$ .

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Determine which of the following side lengths cannot represent a right triangle: $6,8,10$ ; $7,24,25$ ; $10,24,26$ ; $15,25,35$ .

STEP 1

STEP 2

Check option A: $6, 8, 10$ .
We will assume the longest side is the hypotenuse and apply the Pythagorean theorem.
$a^2 + b^2 = c^2$
$6^2 + 8^2 = 10^2$

STEP 3

Calculate the squares of the sides for option A.
$36 + 64 = 100$

STEP 4

Check if the sum of the squares of the two shorter sides equals the square of the longest side for option A.
$36 + 64 = 100$
$100 = 100$
Since the equation holds true, option A can represent a right triangle.

STEP 5

Check option B: $7, 24, 25$ .
Again, we will assume the longest side is the hypotenuse and apply the Pythagorean theorem.
$a^2 + b^2 = c^2$
$7^2 + 24^2 = 25^2$

STEP 6

Calculate the squares of the sides for option B.
$49 + 576 = 625$

STEP 7

Check if the sum of the squares of the two shorter sides equals the square of the longest side for option B.
$49 + 576 = 625$
$625 = 625$
Since the equation holds true, option B can represent a right triangle.

STEP 8

Check option C: $10, 24, 26$ .
Again, we will assume the longest side is the hypotenuse and apply the Pythagorean theorem.
$a^2 + b^2 = c^2$
$10^2 + 24^2 = 26^2$

STEP 9

Calculate the squares of the sides for option C.
$100 + 576 = 676$

STEP 10

STEP 11

Check option D: $15, 25, 35$ .
Again, we will assume the longest side is the hypotenuse and apply the Pythagorean theorem.
$a^2 + b^2 = c^2$
$15^2 + 25^2 = 35^2$

STEP 12

Calculate the squares of the sides for option D.
$225 + 625 = 1225$

STEP 13

STEP 14

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Determine which of the following side lengths cannot represent a right triangle: 6,8,106,8,106,8,10; 7,24,257,24,257,24,25; 10,24,2610,24,2610,24,26; 15,25,3515,25,3515,25,35.

STEP 1

STEP 2

Check option A: 6,8,106, 8, 106,8,10.We will assume the longest side is the hypotenuse and apply the Pythagorean theorem.a2+b2=c2a^2 + b^2 = c^2a2+b2=c262+82=1026^2 + 8^2 = 10^262+82=102

STEP 3

Calculate the squares of the sides for option A.36+64=10036 + 64 = 10036+64=100

STEP 4

Check if the sum of the squares of the two shorter sides equals the square of the longest side for option A.36+64=10036 + 64 = 10036+64=100100=100100 = 100100=100Since the equation holds true, option A can represent a right triangle.

STEP 5

Check option B: 7,24,257, 24, 257,24,25.Again, we will assume the longest side is the hypotenuse and apply the Pythagorean theorem.a2+b2=c2a^2 + b^2 = c^2a2+b2=c272+242=2527^2 + 24^2 = 25^272+242=252

STEP 6

Calculate the squares of the sides for option B.49+576=62549 + 576 = 62549+576=625

STEP 7

Check if the sum of the squares of the two shorter sides equals the square of the longest side for option B.49+576=62549 + 576 = 62549+576=625625=625625 = 625625=625Since the equation holds true, option B can represent a right triangle.

STEP 8

Check option C: 10,24,2610, 24, 2610,24,26.Again, we will assume the longest side is the hypotenuse and apply the Pythagorean theorem.a2+b2=c2a^2 + b^2 = c^2a2+b2=c2102+242=26210^2 + 24^2 = 26^2102+242=262

STEP 9

Calculate the squares of the sides for option C.100+576=676100 + 576 = 676100+576=676

STEP 10

STEP 11

Check option D: 15,25,3515, 25, 3515,25,35.Again, we will assume the longest side is the hypotenuse and apply the Pythagorean theorem.a2+b2=c2a^2 + b^2 = c^2a2+b2=c2152+252=35215^2 + 25^2 = 35^2152+252=352

STEP 12

Calculate the squares of the sides for option D.225+625=1225225 + 625 = 1225225+625=1225

STEP 13

STEP 14

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Determine which of the following side lengths cannot represent a right triangle: $6,8,10$ ; $7,24,25$ ; $10,24,26$ ; $15,25,35$ .

Check option A: $6, 8, 10$ .
We will assume the longest side is the hypotenuse and apply the Pythagorean theorem.
$a^2 + b^2 = c^2$
$6^2 + 8^2 = 10^2$

Calculate the squares of the sides for option A.
$36 + 64 = 100$

Check if the sum of the squares of the two shorter sides equals the square of the longest side for option A.
$36 + 64 = 100$
$100 = 100$
Since the equation holds true, option A can represent a right triangle.

Check option B: $7, 24, 25$ .
Again, we will assume the longest side is the hypotenuse and apply the Pythagorean theorem.
$a^2 + b^2 = c^2$
$7^2 + 24^2 = 25^2$

Calculate the squares of the sides for option B.
$49 + 576 = 625$

Check if the sum of the squares of the two shorter sides equals the square of the longest side for option B.
$49 + 576 = 625$
$625 = 625$
Since the equation holds true, option B can represent a right triangle.

Check option C: $10, 24, 26$ .
Again, we will assume the longest side is the hypotenuse and apply the Pythagorean theorem.
$a^2 + b^2 = c^2$
$10^2 + 24^2 = 26^2$

Calculate the squares of the sides for option C.
$100 + 576 = 676$

Check option D: $15, 25, 35$ .
Again, we will assume the longest side is the hypotenuse and apply the Pythagorean theorem.
$a^2 + b^2 = c^2$
$15^2 + 25^2 = 35^2$

Calculate the squares of the sides for option D.
$225 + 625 = 1225$