Using the Pythagorean Theorem: Applying the formula

To find side AB, we will need to use the Pythagorean theorem.

The Pythagorean theorem allows us to find the third side of a right triangle, if we have the other two sides.

You can read all about the theorem here.

Pythagorean theorem:

$A^2+B^2=C^2$

That is, one side squared plus the second side squared equals the third side squared.

We replace the existing data:

$3^2+2^2=AB^2$

$9+4=AB^2$

$13=AB^2$

We find the root:

$\sqrt{13}=AB$

To find the length of the hypotenuse BC in a right-angled triangle where AB and AC are the other two sides, we use the Pythagorean theorem: $c^2 = a^2 + b^2$.

Here, $a = 6 \text{ cm}$ and $b = 8 \text{ cm}$.

Plugging the values into the Pythagorean theorem, we get:

$c^2 = 6^2 + 8^2$.

Calculating further:

$c^2 = 36 + 64$

$c^2 = 100$.

Taking the square root of both sides gives:

$c = 10 \text{ cm}$.

To solve the problem, we'll follow these steps:

• Step 1: Identify the given sides as the legs of the right triangle.
• Step 2: Use the Pythagorean theorem, $a^2 + b^2 = c^2$, where the legs $a$ and $b$ are both 5 units.
• Step 3: Calculate $c$ using the formula and determine AC.

Let's work through the solution in detail:
Step 1: We have a right triangle with sides $AB = 5$ and $BC = 5$. These represent the legs of the triangle.

Step 2: According to the Pythagorean theorem, the hypotenuse $AC = c$ can be calculated as follows:

$a^2 + b^2 = c^2$
Substituting the values $a = 5$ and $b = 5$:
$5^2 + 5^2 = c^2$

Step 3: Calculate:

$25 + 25 = c^2$
$50 = c^2$
To find $c$, take the square root of both sides:
$c = \sqrt{50}$

Therefore, the length of AC is $\sqrt{50}$.

To find the hypotenuse $AC$ in the right triangle $ABC$, we will apply the Pythagorean Theorem. The theorem states that the square of the hypotenuse $c$ (which is the side opposite the right angle), is equal to the sum of the squares of the other two sides $a$ and $b$.

Given that side $AB = 2$ and $BC = 2$, we have:

• $a = 2$
• $b = 2$

According to the Pythagorean Theorem, $c^2 = a^2 + b^2$.

Substituting the given values into the theorem:

$c^2 = 2^2 + 2^2$

$c^2 = 4 + 4$

$c^2 = 8$

To solve for $c$, take the square root of both sides:

$c = \sqrt{8}$

Therefore, the length of the hypotenuse $AC$ is $\sqrt{8}$.

Among the given choices, the correct answer is choice 2: $\sqrt8$.

To find the length of AC, the hypotenuse of the right triangle ABC, we will apply the Pythagorean Theorem:

The Pythagorean Theorem states:

$a^2 + b^2 = c^2$,

where $a$ and $b$ are the legs ($AB$ and $BC$), and $c$ is the hypotenuse ($AC$).
Given: $AB = 7$ and $BC = 1$.

Substituting the given values into the theorem:

$(7)^2 + (1)^2 = AC^2$.

Calculating the squares:

$49 + 1 = AC^2$.

Simplifying the equation:

$50 = AC^2$.

To find $AC$, take the square root of both sides:

$AC = \sqrt{50}$.

Therefore, the length of AC is $\sqrt{50}$.

To solve the problem of finding the hypotenuse AC in right triangle ABC, we will use the Pythagorean theorem:

• Step 1: Identify leg lengths. According to the problem, leg AB is $11$, and leg BC is $10$.
• Step 2: Apply the Pythagorean theorem: $a^2 + b^2 = c^2$.
• Step 3: Substitute the known values: $11^2 + 10^2 = AC^2$.
• Step 4: Calculate: $121 + 100 = AC^2$.
• Step 5: Simplify: $221 = AC^2$.
• Step 6: Take the square root of both sides to solve for AC: $AC = \sqrt{221}$.

Therefore, the length of AC is $\sqrt{221}$.

To solve this problem, we'll apply the Pythagorean theorem to find the hypotenuse $AC$ of the right triangle $\triangle ABC$.

According to the Pythagorean theorem:
$a^2 + b^2 = c^2$

Here, $a = AB = 6$ and $b = BC = 18$ are the legs of the triangle, and $c = AC$ is the hypotenuse.

Substitute these values into the formula:
$6^2 + 18^2 = AC^2$

Calculate each square:
$6^2 = 36$
$18^2 = 324$

Add the squares of the legs:
$36 + 324 = 360$

Thus,
$AC^2 = 360$
Taking the square root of both sides gives:
$AC = \sqrt{360}$

Therefore, the length of $AC$ is $\sqrt{360}$.

To solve for the hypotenuse $AC$ in triangle $\triangle ABC$, we'll use the Pythagorean theorem:

• Step 1: Identify sides: In the isosceles right triangle, let $AB = 80$, $BC = 20$. These are the two legs.
• Step 2: Substitute into the Pythagorean theorem: $AB^2 + BC^2 = AC^2$.
• Step 3: Perform calculations:

Substitute the values into the equation:
$80^2 + 20^2 = AC^2$.

Calculate each square value:
$6400 + 400 = AC^2$.

Combine the values:
$6800 = AC^2$.

To solve for $AC$, take the square root of both sides:
$AC = \sqrt{6800}$.

Therefore, the solution for the length of $AC$ is $\sqrt{6800}$.

To solve this problem, we will use the Pythagorean Theorem. It states that for a right triangle with sides $a$ and $b$, and hypotenuse $c$, the relationship is given by:

$a^2 + b^2 = c^2$

In triangle ABC, let $AB = a = 6$ and $BC = b = 5$. We need to find the length of the hypotenuse $AC = c$.

Applying the Pythagorean Theorem, we have:

$6^2 + 5^2 = c^2$

$36 + 25 = c^2$

$61 = c^2$

To find $c$, we take the square root of both sides:

$c = \sqrt{61}$

Thus, the length of AC is $\sqrt{61}$.

To solve for the length of side AC of triangle ABC, we will use the Pythagorean theorem as follows:

• Step 1: Identify the sides - Side AB is 8 units and side BC is 3 units.
• Step 2: Apply the Pythagorean theorem - $a^2 + b^2 = c^2$ where $c$ is the hypotenuse.
• Step 3: Substitute the lengths into the formula - $3^2 + 8^2 = AC^2$.
• Step 4: Calculate - $9 + 64 = AC^2$.
• Step 5: Simplify - $73 = AC^2$.
• Step 6: Solve for AC - $AC = \sqrt{73}$.

Therefore, the length of side AC in triangle ABC is $\sqrt{73}$.

To find the length of side $AC$ in the right triangle $\triangle ABC$, we'll use the Pythagorean Theorem:

• Step 1: Identify the given sides. Side $AB = 3$, and hypotenuse $BC = 10$.
• Step 2: Apply the Pythagorean Theorem: $AC^2 = AB^2 + BC^2$.
• Step 3: Substitute the known values: $AC^2 = 3^2 + 10^2 = 9 + 100 = 109$.
• Step 4: Solve for $AC$: $AC = \sqrt{109}$.

Upon calculating, we find that the length of $AC$ is $\sqrt{109}$.

This length corresponds to choice 3 from the provided options.

Thus, the solution to this problem is $AC = \sqrt{109}$.

To solve this problem, we'll use the Pythagorean Theorem.

• Step 1: Identify the sides: $AB = 8$, $BC = 3$.
• Step 2: Use the formula: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse (AC).
• Step 3: Plug in the values: $3^2 + 8^2 = AC^2$.
• Step 4: Calculate: $9 + 64 = AC^2$.
• Step 5: Simplify: $73 = AC^2$.
• Step 6: Solve for $AC$: $AC = \sqrt{73}$.

Thus, the length of AC is $\sqrt{73}$.

To find the length of the hypotenuse $m$ in right triangle $\triangle ABC$, we will use the Pythagorean Theorem: $a^2 + b^2 = c^2$, where $a$ and $b$ are the lengths of the legs and $c$ is the hypotenuse.

• Step 1: Identify the lengths of the legs. Here, $a = 2$ and $b = 1$.
• Step 2: Apply the Pythagorean Theorem: $(2)^2 + (1)^2 = m^2$.
• Step 3: Calculate: $4 + 1 = 5$.
• Step 4: Solve for $m$ by taking the square root: $m = \sqrt{5}$.

Therefore, the length of the hypotenuse $m$ is $\sqrt{5}$.

This matches the provided answer choice: $\sqrt{5}$ (choice 4).

To solve the exercise, we have to use the Pythagorean theorem:

A²+B²=C²

We replace the data we have:

3²+4²=C²

9+16=C²

25=C²

5=C

We use the Pythagorean theorem

$AC^2+AB^2=BC^2$

We insert the known data:

$3^2+4^2=BC^2$

$9+16=BC^2$

$25=BC^2$

We extract the root:

$\sqrt{25}=BC$

$5=BC$