in figure pqrs is a parallelogram and key is midpoint of the sides PQ prove that area of triangle PQ is equals to half of area of triangle skr
To prove that the area of triangle PQ is equal to half the area of triangle SKR, we can use the concept of midpoint of a line segment.
Let's denote the given points as follows:
P, Q, R, and S are the vertices of the parallelogram.
K is the midpoint of the line segment PQ.
To prove that the area of triangle PQ is half the area of triangle SKR, we need to show that these two triangles share the same base and have heights that are in a 1:2 ratio.
Here's how we can prove it:
Step 1: Draw the diagonal SR of the parallelogram PQRS. This diagonal divides the parallelogram into two congruent triangles, namely triangle SKR and triangle PAQ.
Step 2: Since K is the midpoint of the line segment PQ, we can conclude that the line segment SK is congruent to the line segment KR. This is because in a parallelogram, the opposite sides are equal in length.
Step 3: Now, let's consider the heights of the two triangles:
- The height of triangle PQ is equal to the distance between the line segments SK and KR.
- The height of triangle SKR is equal to the distance between the line segment PQ and the line segment SR.
Step 4: Because K is the midpoint of PQ, its height is half the height of PQ. Thus, the height of triangle SKR is twice the height of triangle PQ.
Step 5: We can now calculate the areas of the two triangles using the formula: Area = (base * height) / 2.
Let's denote the base of each triangle as b:
- The base of triangle PQ is equal to the length of PQ.
- The base of triangle SKR is equal to the length of SR.
Since SR is a diagonal of the parallelogram PQRS, its length is equal to the length of PQ. Therefore, the bases of the two triangles are equal.
Step 6: Applying the formula for area, we get:
- Area of triangle PQ = (b * h_PQ) / 2
- Area of triangle SKR = (b * h_SKR) / 2
Since the bases (b) of the triangles are equal, we can rewrite the equations:
- Area of triangle PQ = (b * h_PQ) / 2
- Area of triangle SKR = (b * (2 * h_PQ)) / 2
Simplifying the equation for the area of triangle SKR:
- Area of triangle SKR = b * h_PQ
Here we can see that the area of triangle SKR is twice the area of triangle PQ, as the height of triangle SKR (2 * h_PQ) is twice the height of triangle PQ (h_PQ).
Therefore, we have proved that the area of triangle PQ is equal to half the area of triangle SKR in this given parallelogram.
I would gladly try to answer your question if I could interpret what you mean.
"key is midpoint" --> nonsense
"of the sides PQ" ----> no other side given
"area of triangle PQ" -- PQ is a line segment, not a triangle.
Is your "caps lock" on your keyboard frozen ?