if a right triangle has legs measuring 10 cm and 24 cm, how long is the hypotenuse? (1 point)

Answer 1

To find the length of the hypotenuse in a right triangle, we can use the Pythagorean theorem. The theorem states that in a right triangle:

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

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

Given:

• \(a = 10 , \text{cm}\)
• \(b = 24 , \text{cm}\)

Plugging in the values:

\[ c^2 = 10^2 + 24^2 \]

Calculating it step-by-step:

\[ c^2 = 100 + 576 \] \[ c^2 = 676 \]

Now, take the square root of both sides:

\[ c = \sqrt{676} \] \[ c = 26 , \text{cm} \]

Therefore, the length of the hypotenuse is 26 cm.

Answer 2

Camila needs to fix the window on the front of her house. The bottom of the window is 16 ft. from the ground. She only has a 20 ft ladder. How far from the base of the house should she set up the ladder so that the top of the ladder is exactly at the bottom of the window? (1 point)

Answer 3

To find out how far from the base of the house Camila should place the ladder, we can use the Pythagorean theorem. In this scenario:

• The height from the ground to the bottom of the window is one leg of the right triangle (16 ft).
• The length of the ladder is the hypotenuse (20 ft).
• The distance from the base of the house to the bottom of the ladder is the other leg of the triangle, which we need to find.

Let \( x \) be the distance from the base of the house to the bottom of the ladder. According to the Pythagorean theorem:

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

Substituting into the equation, where \( l = 20 , \text{ft} \), \( a = 16 , \text{ft} \), and \( b = x \):

\[ 20^2 = 16^2 + x^2 \]

Calculating the squares:

\[ 400 = 256 + x^2 \]

Now, subtract 256 from both sides to solve for \( x^2 \):

\[ 400 - 256 = x^2 \] \[ 144 = x^2 \]

Taking the square root of both sides:

\[ x = \sqrt{144} \] \[ x = 12 , \text{ft} \]

Therefore, Camila should set up the ladder 12 feet from the base of the house.

Answer 4

Sylvia likes to walk to work when the weather is nice. She works at the bank at the corner of 41st street and Edison avenue. Her house is at the corner of 42nd street and Johnson avenue. She can walk down Johnson avenue and turn right on 41st street, or she can walk through the park. The distance from 42nd street to 41st street is 80 yards, and it is 60 yards from Johnson Avenue to Edison avenue.
Assuming the street intersections are 90 degree angles, how much shorter is it for Sylvia to walk through the park (1 point)

Answer 5

To determine how much shorter it is for Sylvia to walk through the park compared to walking down Johnson Avenue and then turning right, we need to calculate both distances.

1. Distance walking down Johnson Avenue and turning right on 41st street:

• The distance from 42nd street to 41st street (vertical distance) is 80 yards.
• The distance from Johnson Avenue to Edison Avenue (horizontal distance) is 60 yards.

Total distance walking this route: \[ \text{Distance} = 80 , \text{yards} + 60 , \text{yards} = 140 , \text{yards} \]

2. Distance walking through the park:

Since the park is a straight line connecting the two points (from her house at 42nd street and Johnson Avenue to the bank at 41st street and Edison Avenue), we can use the Pythagorean theorem to find the distance.

• The vertical distance (change in street) is 80 yards (from 42nd street to 41st street).
• The horizontal distance (change in avenue) is 60 yards (from Johnson Avenue to Edison Avenue).

Calculating the distance through the park: \[ d = \sqrt{(80 , \text{yards})^2 + (60 , \text{yards})^2} \] \[ d = \sqrt{6400 + 3600} \] \[ d = \sqrt{10000} \] \[ d = 100 , \text{yards} \]

3. Difference in distances:

Now we can find out how much shorter the park route is:

\[ \text{Difference} = \text{Distance via Johnson Avenue} - \text{Distance through park} \] \[ \text{Difference} = 140 , \text{yards} - 100 , \text{yards} \] \[ \text{Difference} = 40 , \text{yards} \]

Therefore, walking through the park is 40 yards shorter than walking down Johnson Avenue and turning right.