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📐 Class X – Geometry

Similarity, Area & Proportions

A structured problem set for ICSE / CBSE board practice

📅 Updated 2026 📖 9 Problems ⏱️ ~60 min practice 🏷️ Triangles • Parallelograms • Trapezium

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📚 Quick Recap – Geometry Essentials

1. Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional. The main criteria are:

• AA (Angle‑Angle): Two angles of one triangle equal to two angles of another.
• SAS (Side‑Angle‑Side): One angle equal and the sides including it are proportional.
• SSS (Side‑Side‑Side): All three sides are proportional.

2. Basic Proportionality Theorem (Thales)

If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those sides in the same ratio.

3. Mid‑Point Theorem

The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

4. Properties of Parallelograms

• Opposite sides are equal and parallel.
• Diagonals bisect each other.
• Opposite angles are equal.

5. Area Ratios of Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

💡 Pro‑tip: Always look for parallel lines – they are the key to setting up similarity and proportional segments.

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Geometry Problem for Secondary Students

ABC is an isosceles triangle whose ∠C is right angle. If D is any point on AB, then prove that, ABC is an isosceles triangle whose ∠C is right angle. If D is any point on AB, then let us prove that, AD² + DB² = 2CD². ​

Problem from Geometry - Circles

In the figure given below, ABD is a right-angled triangle at B. Taking AB as diameter, a circle has been drawn intersecting AD at F. Prove that the tangent drawn at point F bisects BD.

Solution:

Geometry Solved Problems : Circles

Solved problems on circles for class X. Get solutions of problems shown in the post and many more. Visit the link Solved Problems: Circles

Gergonne triangle

The Gergonne triangle of triangle ABC is defined by the three touchpoints of the incircle on the three sides. This Gergonne triangle, is also known as the contact triangle or intouch triangle of triangle ABC.

#Solution: https://t.me/PrimeMaths/47

#Geometry #PrimeMaths

Problem based on Angle bisector

If AD is the angle bisector of ∠BAC in the triangle ABC. Show that AB.AC=BD.DC+AD^2.

#Solution https://t.me/PrimeMaths/43

#Geometry #PrimeMaths #Angle #Bisectors