Using the Pythagorean Theorem: Isosceles triangle

To solve this problem, we need to determine the length of side $X$ in the right isosceles triangle $\triangle ABC$ with the hypotenuse $\sqrt{50}$.

• Step 1: Recall the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse. For this isosceles case, where both legs $X$ are equal:
$X^2 + X^2 = (\sqrt{50})^2$
• Step 2: Simplify the equation:
$2X^2 = 50$
• Step 3: Solve for $X^2$ by dividing each term:
$X^2 = \frac{50}{2} = 25$
• Step 4: Take the square root of both sides to solve for $X$:
$X = \sqrt{25} = 5$

Therefore, the length of $X$ in the right isosceles triangle is $5$.

We use the Pythagorean theorem as shown below:

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

Since the triangles are isosceles, the theorem can be written as follows:

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

We then insert the known data:

$2AC^2=(8\sqrt{2})^2=64\times2$

Finally we reduce the 2 and extract the root:

$AC=\sqrt{64}=8$

$BC=AC=8$

Since the height divides the base into two equal parts, each part will be called X

We begin by calculating X:$120:2=60$

We then are able to calculate the height of the pyramid using the Pythagorean theorem:

$X^2+H^2=150^2$

We insert the corresponding data:

$60^2+h^2=150^2$

Finally we extract the root: $h=\sqrt{150^2-60^2}=\sqrt{22500-3600}=\sqrt{18900}$

$h=30\sqrt{21}$

To find the missing side, we use the Pythagorean theorem in the upper triangle.

Since the triangle is isosceles, we know that the length of both sides is 7.

Therefore, we apply Pythagoras$A^2+B^2=C^2$$7^2+7^2=49+49=98$

Therefore, the area of the missing side is:$\sqrt{98}$

The area of a rectangle is the multiplication of the sides, therefore:

$\sqrt{98}\times10=98.99\approx99$