If a right triangle has a leg with length 12 and the hypotenuse is 16, how long is the other leg? Round to the nearest tenth, if necessary box

Answer 1

To find the length of the other leg in a right triangle, we can use the Pythagorean theorem, which states that:

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

where \(a\) and \(b\) are the legs of the triangle and \(c\) is the hypotenuse.

In this problem, we have one leg \(a = 12\) and the hypotenuse \(c = 16\). We need to find the length of the other leg \(b\).

Substituting the known values into the Pythagorean theorem:

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

Calculating the squares:

\[ 144 + b^2 = 256 \]

Next, we isolate \(b^2\):

\[ b^2 = 256 - 144 \] \[ b^2 = 112 \]

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

\[ b = \sqrt{112} \]

To simplify \(\sqrt{112}\):

\[ \sqrt{112} = \sqrt{16 \cdot 7} = \sqrt{16} \cdot \sqrt{7} = 4\sqrt{7} \]

Calculating \(4\sqrt{7}\):

Using the approximate value \(\sqrt{7} \approx 2.64575\),

\[ 4\sqrt{7} \approx 4 \cdot 2.64575 \approx 10.583 \]

Rounding it to the nearest tenth:

\[ b \approx 10.6 \]

So, the length of the other leg is:

\[ \boxed{10.6} \]

Answer 2

If a right triangle has a leg with length 12 and the hypotenuse is 16, how long is the other leg? Round to the nearest tenth, if necessary.(1 point)

Answer 3

To find the length of the other leg in a right triangle, we can use the Pythagorean theorem:

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

where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse.

Given:

• One leg \(a = 12\)
• Hypotenuse \(c = 16\)

We need to find the other leg \(b\).

Substituting the given values into the Pythagorean theorem:

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

Calculating the squares:

\[ 144 + b^2 = 256 \]

Now, isolate \(b^2\):

\[ b^2 = 256 - 144 \] \[ b^2 = 112 \]

Next, take the square root of both sides:

\[ b = \sqrt{112} \]

We can simplify \(\sqrt{112}\):

\[ \sqrt{112} = \sqrt{16 \cdot 7} = \sqrt{16} \cdot \sqrt{7} = 4\sqrt{7} \]

Using the approximate value \(\sqrt{7} \approx 2.64575\):

\[ b \approx 4 \cdot 2.64575 \approx 10.583 \]

Rounding to the nearest tenth:

\[ b \approx 10.6 \]

Thus, the length of the other leg is:

\[ \boxed{10.6} \]

Answer 4

Use the image to answer the question.

An illustration shows a square with smaller squares representing baseball plates on the top left, top right, and bottom right vertices. The bottom left vertex has a pentagon that represents home plate. A stick figure is located on the top of the square and the measurement from the right vertex to the stick figure is labeled 30 feet. A dashed diagonal line runs from the stick figure to the inner right side of the square, forming the hypotenuse of a triangle. The top right side of the square forms the right angle of the triangle and is marked with a right angle symbol. The outer right side of the square is labeled 90 feet.

A shortstop is standing in the base path between second and third base when she fields the ball. She is 30 feet from second base and needs to make the throw to first base. Assuming the field is square, how far does she need to throw the ball? Round your answer to the nearest tenth.

(1 point)
feet
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Answer 5

To determine how far the shortstop needs to throw the ball from her position to first base, we can visualize the field as a square and use the given distances to apply the Pythagorean theorem.

1. Understand the Setup:

   • The square field has sides measuring 90 feet (since the outer right side of the square is labeled 90 feet).
   • The shortstop is located 30 feet away from second base and needs to throw the ball to first base.

2. Positioning the Bases:

   • Let's assign the corners of the square:
     • Bottom left corner (home plate)
     • Bottom right corner (first base)
     • Top right corner (second base)
     • Top left corner (third base)

3. Distance from Second Base to Shortstop:

   • If the shortstop is 30 feet from second base, she is positioned on the diagonal line between second base and third base.

4. Calculate the throw distance:

   • The position of the shortstop effectively forms a right triangle with the following sides:
     • Horizontal leg from second base to first base (90 feet)
     • Vertical leg represents the distance she is from second base towards third base (30 feet).

Using the Pythagorean theorem:

\[ d = \sqrt{(90)^2 + (30)^2} \]

Calculating the squares:

\[ d = \sqrt{8100 + 900} \] \[ d = \sqrt{9000} \] \[ d \approx 94.87 \text{ feet} \]

Rounding to the nearest tenth:

\[ d \approx 94.9 \text{ feet} \]

Therefore, the distance the shortstop needs to throw the ball to first base is:

\[ \boxed{94.9} \text{ feet} \]

Answer 6

top is 30 ft and the botoom left is 90ft

Answer 7

Thank you for the clarification regarding the positions and distances.

Based on your description, here's an updated understanding:

• The distance from the shortstop to second base is vertically
• The distance from home plate (the bottom left corner of the square) to first base (the bottom right corner of the square) is 90 feet horizontally.
• The vertical distance from the shortstop (30 feet from second base) goes down to the side of the square.

Since the bases are positioned in a square formation, we need to determine the throw distance from the shortstop's position to first base.

1. Positioning:

   • Shortstop is positioned vertically down from second base, 30 feet.
   • The distance from second base to first base horizontally is still 90 feet.

2. Form the Triangle:

   • The horizontal distance (\(b\)) from second base to first base is 90 feet.
   • The vertical distance (\(a\)) from the shortstop to the line between second base and home plate is 30 feet.

3. Applying the Pythagorean Theorem: To find the distance (\(d\)) that she needs to throw the ball, we can calculate:

\[ d = \sqrt{(90)^2 + (30)^2} \]

Calculating the squares:

\[ d = \sqrt{8100 + 900} \] \[ d = \sqrt{9000} \] \[ d \approx 94.87 \]

Rounding to the nearest tenth:

\[ d \approx 94.9 \text{ feet} \]

Therefore, the distance the shortstop needs to throw the ball is:

\[ \boxed{94.9} \text{ feet} \]

If there's any more specific information or changes, please let me know!

Answer 8

A triangle has sides a
, b
, and c
. Use the Converse of the Pythagorean Theorem to determine whether a right triangle can be formed if a=3
, b=4
, and c=5
.

Option A: Yes, a right triangle can be formed.

Option B: No, a right triangle cannot be formed.

Choose 1 for Option A or 2 for Option B.

(1 point)

Answer 9

To determine whether a right triangle can be formed using the given side lengths \(a = 3\), \(b = 4\), and \(c = 5\), we can use the Converse of the Pythagorean Theorem, which states that if \(c\) is the longest side and satisfies the equation \(a^2 + b^2 = c^2\), then a right triangle can be formed.

Let's check this:

1. Identify \(c\):

   • Here, \(c = 5\) (the longest side).

2. Check the condition using the lengths:

   • Calculate \(a^2 + b^2\): \[ a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25 \]
   • Calculate \(c^2\): \[ c^2 = 5^2 = 25 \]

3. Compare:

   • Since \(a^2 + b^2 = c^2\) (i.e., \(25 = 25\)), this confirms that the condition for a right triangle is satisfied.

Thus, the answer is:

Option A: Yes, a right triangle can be formed.

So you should choose 1 for Option A.

Answer 10

u sure

Answer 11

Yes, I am sure. Let's break it down once more clearly to confirm the conclusion:

You have:

• \( a = 3 \)
• \( b = 4 \)
• \( c = 5 \)

To determine if these sides can form a right triangle, we use the Converse of the Pythagorean Theorem.

According to the theorem:

• In a right triangle, if \( c \) is the hypotenuse (the longest side), then the following relationship must hold:

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

Step-by-step Verification:

1. Identify the sides:

   • \( c = 5 \) (longest side)
   • \( a = 3 \)
   • \( b = 4 \)

2. Calculate \( a^2 + b^2 \): \[ a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25 \]

3. Calculate \( c^2 \): \[ c^2 = 5^2 = 25 \]

4. Comparison:

   • Since \( a^2 + b^2 = c^2 \) (i.e., \( 25 = 25 \)), the condition for forming a right triangle is satisfied.

Conclusion:

Thus, with \( a = 3 \), \( b = 4 \), and \( c = 5 \), a right triangle can be formed.

Therefore, the correct answer is: Option A: Yes, a right triangle can be formed.

You should choose 1 for Option A.