Geometry: Midpoints and Perimeter

In triangle ABC, points D and E are the midpoints of sides AC and BC respectively. If AB = 8 cm, CA = 10 cm, and BC = 9 cm, what is the perimeter of triangle CDE?

Given a triangle ABC where D and E are midpoints of sides AC and BC, and AB = 8 cm. What is the length of segment DE?

In quadrilateral ABCD, points P, Q, R, and S are the midpoints of sides AB, BC, CD, and DA, respectively. If diagonals AC = 8 cm and BD = 12 cm, what is the perimeter of quadrilateral PQRS?

In quadrilateral ABCD, where P, Q, R, and S are midpoints of the sides, what shape does PQRS form?

If the perimeter of $\triangle ABC$ is 34 cm and D, E, and F are the midpoints of sides AB, BC, and CA respectively, what is the perimeter of $\triangle DEF$?

Given points P, Q, R, and S are the midpoints of quadrilateral ABCD. If AC = 10 cm and BD = 14 cm, find the perimeter of parallelogram PQRS.

In $\triangle ABC$, D and E are midpoints of AC and BC. AB = 14 cm, AC = 10 cm, BC = 12 cm. Find the perimeter of the quadrilateral ABDE.

Consider quadrilateral ABCD with midpoints P, Q, R, and S on sides AB, BC, CD, and DA, respectively. If PQRS is a rectangle, which of the following is necessarily true about the diagonals of ABCD?

Suppose P, Q, R, and S are midpoints of the sides of square ABCD. What is the ratio of the area of square ABCD to the area of square PQRS?

In triangle ABC, D is the midpoint of AC and E is the midpoint of BC. If the area of triangle ABC is 28 cm$^2$, what is the area of triangle CDE?

If connecting the midpoints of any quadrilateral always results in a parallelogram, what conclusion can be drawn about the properties of the original quadrilateral?

Suppose the vertices of a quadrilateral ABCD are A(0, 0), B(6, 0), C(8, 4), and D(2, 4). Find the coordinates of the midpoint of the quadrilateral formed by connecting the midpoints of the sides of ABCD.

In $\triangle ABC$, D and E are the midpoints of sides AB and AC respectively. If DE is 6 cm, and DE is parallel to BC, what is the length of BC?

In trapezoid ABCD, where AB is parallel to CD, M and N are the midpoints of AD and BC respectively. If AB = 10 cm and CD = 6 cm, what is the length of MN?

Given a quadrilateral ABCD, the midpoints of the sides are connected to form parallelogram PQRS. What condition is needed for PQRS to be a rhombus?

In a parallelogram ABCD, E, F, G, and H are the midpoints of AB, BC, CD, and DA, respectively. What is the ratio of the area of parallelogram EFGH to the area of parallelogram ABCD?

If the area of a square is 100 cm$^2$, and we connect the midpoints of its sides to form a new square, what is the area of the inner square?

If points P, Q, R, and S are midpoints of sides AB, BC, CD, and DA of a quadrilateral ABCD, and PQRS is a square, which of these statements is true?

In triangle ABC, D and E are midpoints of AB and AC, respectively. If BC = 18 cm, what is the length of DE?