In a square ABCD, the diagonals meet at point O. The `DeltaAOB` is

Options

A. isosceles right triangle
B. equilateral triangle
C. isosceles triangle but not right triangle
D. scalene right triangle.

Answer

To solve the problem, we need to analyze the triangle AOB formed by the diagonals of square ABCD. Here's a step-by-step solution:

### Step 1: Draw the Square
Draw a square ABCD. Label the vertices as A, B, C, and D in a clockwise manner.

**Hint:** Remember that all sides of a square are equal, and all angles are right angles (90 degrees).

### Step 2: Draw the Diagonals
Draw the diagonals AC and BD. They will intersect at point O, which is the center of the square.

**Hint:** The diagonals of a square bisect each other and are equal in length.

### Step 3: Identify the Angles
In triangle AOB, we need to identify the angles. Since ABCD is a square, we know that:
- Angle AOB is formed by the diagonals.
- The diagonals of a square intersect at 90 degrees.

**Hint:** The diagonals of a square create four right triangles within the square.

### Step 4: Determine the Angles of Triangle AOB
Since the diagonals intersect at point O and form right angles, we can conclude:
- Angle AOB = 90 degrees
- Angles OAB and OBA are equal because AO = OB (sides of the triangle AOB).

**Hint:** In an isosceles triangle, the angles opposite the equal sides are equal.

### Step 5: Calculate the Angles
Let the angles OAB and OBA both be x. Since the sum of angles in a triangle is 180 degrees, we have:
- Angle AOB + Angle OAB + Angle OBA = 180 degrees
- 90 + x + x = 180
- 90 + 2x = 180
- 2x = 90
- x = 45 degrees

Thus, Angle OAB = Angle OBA = 45 degrees.

**Hint:** Use the property that the sum of angles in a triangle equals 180 degrees to find the unknown angles.

### Step 6: Classify Triangle AOB
Now that we have the angles:
- Angle AOB = 90 degrees
- Angle OAB = 45 degrees
- Angle OBA = 45 degrees

Since two angles are equal and one angle is 90 degrees, triangle AOB is classified as an isosceles right triangle.

**Hint:** A triangle with one 90-degree angle and two equal angles is called an isosceles right triangle.

### Conclusion
The triangle AOB is an isosceles right triangle.

**Final Answer:** Triangle AOB is an isosceles right triangle.