What is the relationship of the sides in this triangle based on the Pythagorean Theorem? (1 point) TRIANGLE: Top left corner is O, Top right corner is N, Bottom left point is M, Left side is n, Right side is o, Top side is m, n has an angle of 28, N has an angle of 62

Answer 1

In triangle \( MON \):

• Label the sides and angles as follows:
  • \( OM = n \) (the left side)
  • \( ON = o \) (the right side)
  • \( MN = m \) (the top side)
  • Angle at \( O \) is \( 28^\circ \)
  • Angle at \( N \) is \( 62^\circ \)

To find the relationship of the sides based on the Pythagorean Theorem, we first observe that angles in a triangle sum up to \( 180^\circ \).

The third angle \( M \) can be calculated as follows:

\[ \angle M = 180^\circ - 28^\circ - 62^\circ = 90^\circ \]

Since angle \( M \) is \( 90^\circ \), triangle \( MON \) is a right triangle.

By the Pythagorean Theorem, the relationship between the sides is given by:

\[ n^2 + m^2 = o^2 \]

In this equation:

• \( n \) is the length of side \( OM \)
• \( m \) is the length of side \( MN \)
• \( o \) is the length of side \( ON \)

Thus, the relationship of the sides in triangle \( MON \) based on the Pythagorean Theorem is \( n^2 + m^2 = o^2 \).

Answer 2

What is the relationship of the sides in this triangle based on the Pythagorean Theorem? (1 point) TRIANGLE: Top left corner is O, Top right corner is N, Bottom left point is M, Left side is n, Right side is o, Top side is m, M has an angle of 28, N has an angle of 62
EQUATION: m^2 + n^2 = ?

Answer 3

In triangle \( MON \):

• Given that angle \( M \) is \( 28^\circ \) and angle \( N \) is \( 62^\circ \), we can find that angle \( O \) is:

\[ \angle O = 180^\circ - 28^\circ - 62^\circ = 90^\circ \]

Since triangle \( MON \) has a right angle at \( O \), it is a right triangle, where:

• \( OM = n \) (the vertical side)
• \( ON = o \) (the hypotenuse)
• \( MN = m \) (the horizontal side)

According to the Pythagorean Theorem, the relationship between the sides in a right triangle is:

\[ m^2 + n^2 = o^2 \]

So, in your equation where \( m^2 + n^2 = ? \), we can fill it in as follows:

\[ m^2 + n^2 = o^2 \]

This means the relationship of the sides in triangle \( MON \) based on the Pythagorean Theorem is \( m^2 + n^2 = o^2 \).

Answer 4

If a triangle has sides measuring 60, 75, and 45, is it a right triangle?(1 point)
Yes, because 5,625 equals 5,625.
Yes, because 3,600 equals 3,600.
No, because 9,225 does not equal 2,025.
No, because 5,625 does not equal 3,600.

Answer 5

To determine if a triangle with sides measuring 60, 75, and 45 is a right triangle, we can apply the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

First, we identify the sides:

• The longest side is \( 75 \) (hypotenuse).
• The other sides are \( 60 \) and \( 45 \).

Now, we compute the squares of the sides:

1. Square of the longest side (hypotenuse): \[ 75^2 = 5625 \]

2. Sum of the squares of the other two sides: \[ 60^2 + 45^2 = 3600 + 2025 = 5625 \]

Now we compare:

• The square of the hypotenuse: \( 5625 \)
• The sum of the squares of the other two sides: \( 5625 \)

Since \( 5625 = 5625 \), the condition of the Pythagorean theorem is satisfied.

Therefore, the triangle is a right triangle, and the correct answer is:

Yes, because 5,625 equals 5,625.

Answer 6

If triangle has sides measuring 50, 40, and 32, is the triangle a right triangle? If it is a right
triangle, answer with the number 90. If it is not a right triangle, answer with the number 0.(1 point)

Answer 7

To determine if a triangle with sides measuring \( 50 \), \( 40 \), and \( 32 \) is a right triangle, we can apply the Pythagorean Theorem, which states that if a triangle is a right triangle, then the square of the length of the longest side (the hypotenuse) must equal the sum of the squares of the other two sides.

Identifying the sides:

• The longest side is \( 50 \).
• The other two sides are \( 40 \) and \( 32 \).

Now we compute the squares of the sides:

1. Square of the longest side (hypotenuse): \[ 50^2 = 2500 \]

2. Sum of the squares of the other two sides: \[ 40^2 + 32^2 = 1600 + 1024 = 2624 \]

Now we compare:

• The square of the hypotenuse: \( 2500 \)
• The sum of the squares of the other two sides: \( 2624 \)

Since \( 2500 \) does not equal \( 2624 \), the triangle is not a right triangle.

Therefore, the answer is 0.

Answer 8

If triangle DEF has sides measuring 50, 40, and 32, is the triangle a right triangle? If it is a right
triangle, answer with the number 90. If it is not a right triangle, answer with the number 0.(1 point)

Answer 9

To determine whether triangle \( DEF \) with sides measuring \( 50 \), \( 40 \), and \( 32 \) is a right triangle, we can use the Pythagorean Theorem.

1. Identify the sides:

   • The longest side is \( 50 \) (potential hypotenuse).
   • The other two sides are \( 40 \) and \( 32 \).

2. Calculate the squares of the sides:

   • Square of the longest side (hypotenuse): \[ 50^2 = 2500 \]
   • Sum of the squares of the other two sides: \[ 40^2 + 32^2 = 1600 + 1024 = 2624 \]

3. Compare the results:

   • \( 2500 \) (hypotenuse) vs. \( 2624 \) (sum of the other two sides).

Since \( 2500 \) does not equal \( 2624 \), triangle \( DEF \) is not a right triangle.

Therefore, the answer is 0.

Answer 10

If a triangle has sides of length 12 and 16, how long is the hypotenuse?(1 point)

Answer 11

To find the length of the hypotenuse in a right triangle with sides measuring \( 12 \) and \( 16 \), we can use the Pythagorean Theorem:

\[ c^2 = a^2 + b^2 \]

where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides.

Plugging in the values:

\[ c^2 = 12^2 + 16^2 \]

Calculating the squares:

\[ c^2 = 144 + 256 \]

Now summing those:

\[ c^2 = 400 \]

To find \( c \), take the square root:

\[ c = \sqrt{400} = 20 \]

Thus, the length of the hypotenuse is 20.