Find the unknown side length in the given triangle. Round to the nearest hundredth

To find the unknown base length in the right triangle, we can use the Pythagorean theorem:

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

where:

• \( a \) is one leg (the perpendicular side)
• \( b \) is the other leg (the base)
• \( c \) is the hypotenuse

In this case:

• \( a = 20 \) (the perpendicular side)
• \( c = 30 \) (the hypotenuse)
• \( b \) is the unknown base length we want to find.

Plugging the known values into the Pythagorean theorem gives:

\[ 20^2 + b^2 = 30^2 \]

Calculating the squares:

\[ 400 + b^2 = 900 \]

Next, we isolate \( b^2 \) by subtracting 400 from both sides:

\[ b^2 = 900 - 400 \]

\[ b^2 = 500 \]

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

\[ b = \sqrt{500} \]

Calculating the square root:

\[ b \approx 22.36 \]

Rounding this to the nearest hundredth gives:

\[ b \approx 22.36 \]

Thus, the unknown base length is approximately 22.36.

use the image to answer the question.

A rectangle is formed from two right triangles. The base of the rectangle measures 80 yards and is labeled Johnson Avenue. The top of the rectangle is labeled Edison Avenue. The left side joining both the avenues is labeled 41st street and measures 60 yards. The right side of the rectangle is labeled 42nd street. A bank is located at the vertex of 41st street and Edison Avenue. The hypotenuse joining the bank and the vertex of Johnson Avenue and 42nd street is labeled park.

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° angles, how much shorter is it for Sylvia to walk through the park?

if sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be __ yards shorter

To find out how much shorter Sylvia's walk is when she goes through the park instead of walking straight down Johnson Avenue and then down 41st Street, we can calculate both distances.

1. Distance walking straight down Johnson Avenue and then down 41st Street:

   • Distance down Johnson Avenue (from 42nd Street to Edison Avenue) = 80 yards
   • Distance down 41st Street (from Edison Avenue to the bank) = 60 yards
   • Total distance walking straight = \( 80 + 60 = 140 \) yards

2. Distance walking through the park:

   • The park forms a right triangle with the following sides:
     • One leg = 80 yards (distance between Johnson Avenue and 42nd Street)
     • Another leg = 60 yards (distance between 42nd Street and Edison Avenue)
   • Using the Pythagorean theorem to calculate the hypotenuse (distance through the park):

   \[ d = \sqrt{(80^2) + (60^2)} \] \[ d = \sqrt{6400 + 3600} \] \[ d = \sqrt{10000} \] \[ d = 100 \text{ yards} \]

3. Calculating the difference in distances:

   • Total distance walking straight = 140 yards
   • Distance walking through the park = 100 yards
   • Difference = \( 140 - 100 = 40 \) yards

Therefore, if Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be 40 yards shorter.

Apply the Pythagorean Theorem to find the unknown side length in the given triangle. Round your answer to the nearest tenth.

A right triangle shows the base with a value of 3, the hypotenuse on the left with a value of 15, and the perpendicular side to the base on the right with a question mark.

216
15.3
234
14.7

To find the unknown side length in the right triangle using the Pythagorean theorem, we can use the following formula:

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

Where:

• \( a \) is one leg (the base)
• \( b \) is the other leg (the perpendicular side)
• \( c \) is the hypotenuse

From the problem:

• The base \( a = 3 \)
• The hypotenuse \( c = 15 \)
• The unknown perpendicular side \( b \) is what we want to find.

Plugging the known values into the Pythagorean theorem gives:

\[ 3^2 + b^2 = 15^2 \]

Calculating the squares:

\[ 9 + b^2 = 225 \]

Next, isolate \( b^2 \) by subtracting 9 from both sides:

\[ b^2 = 225 - 9 \]

\[ b^2 = 216 \]

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

\[ b = \sqrt{216} \]

Calculating \( \sqrt{216} \):

\[ b \approx 14.7 \]

Thus, rounding to the nearest tenth gives:

\[ b \approx 14.7 \]

Therefore, the unknown side length is 14.7.