Question

What is the missing length?

Answer 1

Using the Pythagorean theorem, we know that in a right-angle triangle, we have:

a^2 + b^2 = c^2

where c is the hypotenuse.

From the given information, we have:

ab = 11 - x

a = x

b = 11

c = 11 (given)

We can use the given values to eliminate a or b from the Pythagorean theorem:

a^2 + (11 - x)^2 = 11^2

Expanding and simplifying, we get:

a^2 + 121 - 22x + x^2 = 121

a^2 + x^2 - 22x = 0

Substituting a = x into the above equation, we get:

x^2 + x^2 - 22x = 0

2x^2 - 22x = 0

2x(x - 11) = 0

So, either x = 0 or x - 11 = 0.

Since x cannot be zero (as it represents a length), we have x - 11 = 0.

Therefore, x = 11.

Hence, the value of x in its simplest form with a rational denominator is 11/1 or just 11.

Answer 2

`x=11√(2)`

Explanation:

The given right triangle is an isosceles right triangle since its legs are equal in length (denoted by the tick marks).

Side x is the hypotenuse of the isosceles right triangle.

Given both legs are 11 units in length, we can use Pythagoras Theorem to calculate the length of the hypotenuse.

`\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}`

As a and b are the legs, and c is the hypotenuse, substitute the following values into the formula and solve for x:

• a = 11
• b = 11
• c = x

Therefore:

`\implies 11^2+11^2=x^2`

`\implies 121+121=x^2`

`\implies 242=x^2`

`\implies x^2=242`

`\implies √(x^2)=√(242)`

`\implies x=√(242)`

To simplify the radical, rewrite it as a product of prime numbers:

`\implies x=√(11^2 \cdot 2)`

`\textsf{Apply the radical rule:} \quad √(ab)=√(a)√(b)`

`\implies x=√(11^2){√(2)`

`\textsf{Apply the radical rule:} \quad √(a^2)=a, \quad a \geq 0`

`\implies x=11√(2)`

Therefore, the length of side x in simplest radical form is 11√2.