Question

Solve for x.
8
X
x100
||
+1x
x = [?] √
I
4
Enter answer in
simplest form.
Submit

Answer 1

Answer:

$\sf x=4\sqrt{2}$

Step-by-step explanation:

To find the value of x, we can use the Geometric Mean Theorem (Altitude Rule).

Geometric Mean Theorem (Altitude Rule)

The altitude drawn from the vertex of the right angle perpendicular to the hypotenuse separates the hypotenuse into two segments. The ratio of the altitude to one segment is equal to the ratio of the other segment to the altitude:

$\boxed{\sf \dfrac{Altitude}{Segment\:1}=\dfrac{Segment\:2}{Altitude}}$

In this case:

• Altitude = x
• Segment 1 = 8
• Segment 2 = 4

Substitute the values into the formula:

$\sf \dfrac{x}{8}=\dfrac{4}{x}$

Cross multiply:

$\sf x \cdot x=4 \cdot 8$

$\sf x^2=32$

Square root both sides:

$\sf \sqrt{x^2}=\sqrt{32}$

$\sf x=\sqrt{32}$

(We take the positive square root of 32 since length cannot be negative).

Rewrite 32 as the product of 4² and 2:

$\sf x=\sqrt{4^2\cdot 2}$

$\textsf{Apply the radical rule:} \quad \sqrt{ab}=\sqrt{\vphantom{b}a}\sqrt{b}$

$\sf x=\sqrt{4^2}\sqrt{2}$

$\textsf{Apply the radical rule:} \quad \sqrt{a^2}=a, \quad a \geq 0$

$\sf x=4\sqrt{2}$

Therefore, the value of x is:

$\Large\boxed{\boxed{\sf x=4\sqrt{2}}}$

Answer 2

Answer:

$x = 4\sqrt{2}$

Step-by-step explanation:

To find the value of x height of the triangle , we can use Altitude rule:

It states that if an altitude is drawn from the right-angled vertex of a right triangle to the hypotenuse, then the length of the altitude is the geometric mean of the lengths of the two segments into which the hypotenuse is divided.

Mathematically:

$\sf \dfrac{Altitude}{Segment \, 1 } = \dfrac{Segment \, 2 }{Altitude}$

In this case, we have the relationship as:

$\dfrac{x}{8} = \dfrac{4}{x}$

This is the expression based on the geometric mean theorem. Now, let's solve for $x$:

Doing cross multiplication:

$x\cdot x = 8 \cdot 4$

$x^2 = 32$

Taking square root on both sides, we get

$x = \sqrt{32}$

$x = \sqrt{4^2 \cdot 2 }$

$x = 4\sqrt{2}$

So, the height of the triangle (x) is $4\sqrt{2}$.