If all of the diagonals are drawn from a vertex of a quadrilateral, how many triangles are formed?
sheesh! - draw one. There's only one diagonal, making 2 triangles.
a quad has only 4 points. pick a vertex. 2 of the other points form the adjoining side, leaving only 1 point left to form a diagonal.
I looked at it a bit different.
Drawing the two diagonals, which intersect at one point, leaves us with 5 points, from which to choose any 3 to form a triangle
We can choose C(5,3) or 10 such triples,
BUT, two of the triplets are the 3 points that form the diagonals, they cannot form a triangle.
So number of triangles = 10-2 = 8
Count them to verify.
To determine the number of triangles formed when all diagonals are drawn from a vertex of a quadrilateral, we need to understand the pattern and relationship among the diagonals.
Let's analyze the situation step by step:
1. Start with a quadrilateral, which has four vertices (A, B, C, D) and four sides (AB, BC, CD, DA).
2. Choose one vertex, let's say A, to draw diagonals from. This means we will draw three diagonals, connecting vertex A to the three non-adjacent vertices (B, C, and D).
3. Each diagonal drawn from vertex A will intersect with the other diagonals and the sides of the quadrilateral.
4. We need to count the number of triangles formed by these diagonals.
Now, let's examine the relationships:
- When the first diagonal is drawn (from A to B), it will intersect all three remaining sides and form three triangles (ABC, ABD, ACD).
- When the second diagonal is drawn (from A to C), it will intersect the previous diagonal (AB) and the last side (CD). Thus, it will form two additional triangles, but one of them (ABC) is already counted, so only one triangle (ACD) is new.
- Finally, when the third diagonal is drawn (from A to D), it will intersect the first two diagonals (AB and AC) and form two more triangles (ABD, ACD). However, one of them (ACD) is already counted, so only one triangle (ABD) is new.
Therefore, the total number of triangles formed by drawing all diagonals from a vertex of a quadrilateral is:
3 (from the first diagonal) + 1 (from the second diagonal) + 1 (from the third diagonal) = 5 triangles.
Hence, five triangles are formed when all diagonals are drawn from a vertex of a quadrilateral.