Solved on Mar 22, 2024

Determine if a triangle with sides of lengths $97$ , $70$ , and $74$ is right, acute, or obtuse using the Pythagorean Theorem.

STEP 1

Assumptions
1. The sides of the triangle are given as 97, 70, and 74.
2. The Pythagorean Theorem states that for 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. If the square of the longest side is greater than the sum of the squares of the other two sides, the triangle is obtuse.
4. If the square of the longest side is less than the sum of the squares of the other two sides, the triangle is acute.

STEP 2

Identify the longest side of the triangle, which could be the hypotenuse if the triangle is right-angled.
$Longest\, side = 97$

STEP 3

Apply the Pythagorean Theorem to check if the triangle is right-angled.
$c^2 = a^2 + b^2$
Where $c$ is the length of the longest side, and $a$ and $b$ are the lengths of the other two sides.

STEP 4

Substitute the given side lengths into the Pythagorean Theorem.
$97^2 = 70^2 + 74^2$

STEP 5

Calculate the square of the longest side.
$97^2 = 9409$

STEP 6

Calculate the sum of the squares of the other two sides.
$70^2 + 74^2 = 4900 + 5476$

STEP 7

Add the squares of the other two sides.
$4900 + 5476 = 10376$

STEP 8

Compare the square of the longest side with the sum of the squares of the other two sides.
$9409 \, \text{(longest side squared)} \quad \text{vs} \quad 10376 \, \text{(sum of squares of other sides)}$

STEP 9

Determine the type of triangle based on the comparison.
Since $9409 < 10376$ , the square of the longest side is less than the sum of the squares of the other two sides, which means the triangle is acute.
The triangle is acute because the square of the largest side (97) is less than the sum of the squares of the other two sides (70 and 74).

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Determine if a triangle with sides of lengths $97$ , $70$ , and $74$ is right, acute, or obtuse using the Pythagorean Theorem.

STEP 1

STEP 2

Identify the longest side of the triangle, which could be the hypotenuse if the triangle is right-angled.
$Longest\, side = 97$

STEP 3

Apply the Pythagorean Theorem to check if the triangle is right-angled.
$c^2 = a^2 + b^2$
Where $c$ is the length of the longest side, and $a$ and $b$ are the lengths of the other two sides.

STEP 4

Substitute the given side lengths into the Pythagorean Theorem.
$97^2 = 70^2 + 74^2$

STEP 5

Calculate the square of the longest side.
$97^2 = 9409$

STEP 6

Calculate the sum of the squares of the other two sides.
$70^2 + 74^2 = 4900 + 5476$

STEP 7

Add the squares of the other two sides.
$4900 + 5476 = 10376$

STEP 8

Compare the square of the longest side with the sum of the squares of the other two sides.
$9409 \, \text{(longest side squared)} \quad \text{vs} \quad 10376 \, \text{(sum of squares of other sides)}$

STEP 9

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Determine if a triangle with sides of lengths 979797, 707070, and 747474 is right, acute, or obtuse using the Pythagorean Theorem.

STEP 1

STEP 2

Identify the longest side of the triangle, which could be the hypotenuse if the triangle is right-angled.Longest side=97Longest\, side = 97Longestside=97

STEP 3

Apply the Pythagorean Theorem to check if the triangle is right-angled.c2=a2+b2c^2 = a^2 + b^2c2=a2+b2Where c c c is the length of the longest side, and a a a and b b b are the lengths of the other two sides.

STEP 4

Substitute the given side lengths into the Pythagorean Theorem.972=702+74297^2 = 70^2 + 74^2972=702+742

STEP 5

Calculate the square of the longest side.972=940997^2 = 9409972=9409

STEP 6

Calculate the sum of the squares of the other two sides.702+742=4900+547670^2 + 74^2 = 4900 + 5476702+742=4900+5476

STEP 7

Add the squares of the other two sides.4900+5476=103764900 + 5476 = 103764900+5476=10376

STEP 8

STEP 9

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Determine if a triangle with sides of lengths $97$ , $70$ , and $74$ is right, acute, or obtuse using the Pythagorean Theorem.

Identify the longest side of the triangle, which could be the hypotenuse if the triangle is right-angled.
$Longest\, side = 97$

Apply the Pythagorean Theorem to check if the triangle is right-angled.
$c^2 = a^2 + b^2$
Where $c$ is the length of the longest side, and $a$ and $b$ are the lengths of the other two sides.

Substitute the given side lengths into the Pythagorean Theorem.
$97^2 = 70^2 + 74^2$

Calculate the square of the longest side.
$97^2 = 9409$

Calculate the sum of the squares of the other two sides.
$70^2 + 74^2 = 4900 + 5476$

Add the squares of the other two sides.
$4900 + 5476 = 10376$