Question

How many right-angled triangles are there in the given image? 

• A. 20
• B. 24
• C. 28
• D. 32

Hint:

To avoid double-counting or missing shapes in geometry counting problems, categorize the shapes you are looking for based on a distinct feature, such as size, orientation, or, in this case, the location of the right angle.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a geometry problem that requires systematically identifying and counting all the right-angled triangles within a complex figure. The figure is a square with its diagonals and medians (lines connecting midpoints of opposite sides) drawn.
Step 2: Key Formula or Approach:
A right-angled triangle has one angle of exactly 90 degrees. In the given figure, 90-degree angles are formed at:

The four corners of the main square.
The center of the square, where the diagonals and medians intersect (diagonals of a square are perpendicular bisectors of each other).
The four midpoints of the sides, where the medians meet the sides.
We will count the triangles based on where their right angle is located.
Step 3: Detailed Explanation and Counting:


Triangles with the right angle at the CENTER of the square:

The two diagonals are perpendicular. This forms 4 right-angled triangles using the main corners (e.g., triangle with vertices: top-left corner, center, top-right corner).
The two medians are also perpendicular. This forms 4 smaller right-angled triangles using the midpoints of the sides (e.g., triangle with vertices: top-midpoint, center, right-midpoint).
Total with right angle at the center = 4 + 4 = 8.
Triangles with the right angle at the MIDPOINTS of the sides:

Consider the top midpoint. The median is perpendicular to the side. This forms two right-angled triangles (top-midpoint, top-left corner, center) and (top-midpoint, top-right corner, center).
Since there are 4 midpoints, this gives 4 x 2 = 8 such triangles.

Triangles with the right angle at the CORNERS of the square:

The four corners of the large square are right angles.
At each corner, there is one small right-angled triangle formed with the adjacent midpoints (e.g., at the top-left corner, the triangle with vertices: top-left corner, top-midpoint, left-midpoint). This gives 4 such triangles.
At each corner, there is one large right-angled triangle formed by the diagonal of the main square (e.g., at the top-left corner, the triangle with vertices: top-left corner, top-right corner, bottom-left corner). This gives 4 such triangles.
Total with right angle at the corners = 4 + 4 = 8.
Step 4: Final Answer:
Summing up all the right-angled triangles from the three categories:
\[ 8 \text{ (from center)} + 8 \text{ (from midpoints)} + 8 \text{ (from corners)} = 24 \] There are a total of 24 right-angled triangles in the image.