Math / Geometry

Question100 pizz B. 10m C 100 45° A abaily

Studdy Solution

STEP 1

1. The triangle is a right triangle inscribed in a circle.
2. The angle at vertex A is $45^\circ$ .
3. The hypotenuse $BC$ is $10$ meters.
4. We need to find the area of the triangle.

STEP 2

1. Identify the type of triangle and use properties of special triangles.
2. Determine the lengths of the legs of the triangle.
3. Calculate the area of the triangle using the formula $\frac{1}{2} \times \text{base} \times \text{height}$ .

STEP 3

Since the triangle is inscribed in a circle and has a right angle, it is a right triangle. The angle at vertex A is $45^\circ$ , which means the other angle in the triangle is also $45^\circ$ because the sum of angles in a triangle is $180^\circ$ . Therefore, this is a 45-45-90 triangle, which is an isosceles right triangle.

STEP 4

In a 45-45-90 triangle, the legs are equal, and the hypotenuse is $l\sqrt{2}$ , where $l$ is the length of each leg. Given the hypotenuse $BC = 10$ meters, we can find the length of each leg using the relationship:
$10 = l\sqrt{2}$

STEP 5

Solve for $l$ :
$l = \frac{10}{\sqrt{2}}$ $l = \frac{10 \times \sqrt{2}}{2}$ $l = 5\sqrt{2}$
Each leg $AB$ and $AC$ is $5\sqrt{2}$ meters.

STEP 6

Calculate the area of the triangle using the formula $\frac{1}{2} \times \text{base} \times \text{height}$ . Here, both the base and height are $5\sqrt{2}$ :
$\text{Area} = \frac{1}{2} \times 5\sqrt{2} \times 5\sqrt{2}$

STEP 7

Simplify and calculate:
$\text{Area} = \frac{1}{2} \times 5\sqrt{2} \times 5\sqrt{2}$ $= \frac{1}{2} \times 25 \times 2$ $= \frac{50}{2}$ $= 25$
The area of the triangle is:
$\boxed{25} \text{ square meters}$

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