ABC is a triangle right angled at B. P,Q,R are the mid points of the sides AC,BC and AB respectively.PR=3 cm,PQ=4 cm. 1) draw a rough figure (5)

II) What is the length of QR?

III) What are the sides of triangle ABC?

IV) Suggest a suitable name to PQBR?

Let's break down the problem step by step.

### I) Drawing a Rough Figure

1. **Draw Triangle ABC**: Since ABC is a right-angled triangle with the right angle at B, draw a right triangle with vertices A, B, and C. Place B at the right angle.
2. **Locate Midpoints**: Identify points P, Q, and R as the midpoints of sides AC, BC, and AB, respectively.
3. **Draw the Medians**: Connect P to R, P to Q, and R to Q.
4. **Label the Given Lengths**: Mark PR as 3 cm and PQ as 4 cm.
5. **Final Touches**: Ensure all points and lines are clearly marked.

### II) Length of QR

We can find the length of \( QR \) using the fact that \( PQR \) is the medial triangle of \( \triangle ABC \).

- The length of \( QR \) can be found using the fact that in a right triangle, the medial triangle formed by connecting the midpoints of the sides has sides that are half the length of the hypotenuse.

Let’s denote the sides of triangle \( ABC \) as follows:
- \( AB = c \)
- \( BC = a \)
- \( AC = b \) (hypotenuse)

Since \( PQR \) is the medial triangle, each side is half the length of the corresponding sides of \( ABC \).

### III) Sides of Triangle ABC

Given \( PR = 3 \) cm and \( PQ = 4 \) cm, these correspond to half the lengths of sides \( AB \) and \( BC \), respectively:

- \( AB = 2 \times PR = 2 \times 3 = 6 \) cm
- \( BC = 2 \times PQ = 2 \times 4 = 8 \) cm
- To find \( AC \), use the Pythagorean theorem in triangle \( ABC \):

\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm}
\]

### IV) Suitable Name for PQBR

Since \( P, Q, R \) are midpoints and form a quadrilateral with point B, this quadrilateral could be referred to as **PQBR Quadrilateral**. However, based on its properties:
- \( PQBR \) is a **trapezium** (specifically a right trapezium), as \( PR \) and \( BQ \) would be parallel, and one angle is \( 90^\circ \).

### Summary

1. **Length of QR**: 5 cm (half of the hypotenuse AC of triangle ABC).
2. **Sides of Triangle ABC**: AB = 6 cm, BC = 8 cm, AC = 10 cm.
3. **Suitable Name for PQBR**: Right Trapezium.

Do you want details or have any questions?

Here are 5 related questions to explore:

1. What is the area of triangle \( ABC \)?
2. How do you calculate the area of trapezium \( PQBR \)?
3. Can you determine the perimeter of triangle \( ABC \)?
4. What is the relationship between the medians and the centroid of triangle \( ABC \)?
5. How would you prove that \( PQBR \) is a trapezium?

**Tip:** The medial triangle of any triangle divides the original triangle into four smaller triangles of equal area.

Explore the properties of triangle ABC, including calculating the length of QR, determining sides AB, BC, and AC using the Pythagorean theorem, and suggesting a suitable name for the PQBR quadrilateral. Dive into geometric concepts and formulas suitable for high school students.

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