What is the length of the hypotenuse? if necessary round to the nearest tenth.

Question

(the opposite/altitude of the triangle is labelled 6 ft., the base/adjacent is labelled 8 ft.)

Answer 1

To find the length of the hypotenuse in a right triangle, you can use the Pythagorean theorem, which states:

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

where:

• \( c \) is the length of the hypotenuse,
• \( a \) is the length of one leg (the opposite side),
• \( b \) is the length of the other leg (the adjacent side).

In this case, you have:

• \( a = 6 \) ft (the opposite side),
• \( b = 8 \) ft (the base or adjacent side).

Now plug in the values:

\[ c^2 = 6^2 + 8^2 \] \[ c^2 = 36 + 64 \] \[ c^2 = 100 \]

Now take the square root of both sides to find \( c \):

\[ c = \sqrt{100} \] \[ c = 10 \text{ ft} \]

Thus, the length of the hypotenuse is 10 ft.

Answer 2

Determine if each of the following statements is always true, sometimes true or never true. a. The sum of the measures of two supplementary angles is 180 degrees. The sum of complimentary angles is 180 degrees. Two adjacent angles are complementary.

always true, sometimes true, never true

Answer 3

Let's analyze each statement one by one:

a. The sum of the measures of two supplementary angles is 180 degrees.

Always true.
Supplementary angles are defined as two angles whose measures add up to 180 degrees.

b. The sum of complementary angles is 180 degrees.

Never true.
Complementary angles are defined as two angles whose measures add up to 90 degrees, not 180 degrees.

c. Two adjacent angles are complementary.

Sometimes true.
Adjacent angles can be complementary (if they add up to 90 degrees) or they can be supplementary (if they add up to 180 degrees) or they can be neither (if they do not add up to either). Therefore, it depends on the specific angles in question.

Summary:

• a. Always true
• b. Never true
• c. Sometimes true