Geometry Maths 2 SSC (English Medium) 10th Standard Board Exam Maharashtra State Board Syllabus 2024-25

1. Ratio of areas of two triangles is equal to the ratio of the products of their bases and corresponding heights.
2. Areas of triangles with equal heights are proportional to their corresponding bases.
3. Areas of triangles with equal bases are proportional to their corresponding heights.

• Theorem: If a line parallel to a side of a triangle intersects the remaining sides in two distinct points, then the line divides the sides in the same proportion.

• Theorem: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

• Theorem: The bisector of an angle of a triangle divides the side opposite to the angle in the ratio of the remaining sides.

• Theorem: The ratio of the intercepts made on a transversal by three parallel lines is equal to the ratio of the corresponding intercepts made on any other transversal by the same parallel lines.

• AAA test for similarity of triangles
• AA test for similarity of triangles
• SAS test of similarity of triangles
• SSS test for similarity of triangles

• Theorem: When two triangles are similar, the ratio of areas of those triangles is equal to the ratio of the squares of their corresponding sides.

• Formula for Pythagorean triplet:

If a, b, c are natural numbers and a > b, then [(a2+ b2),(a2 - b2),(2ab)] is Pythagorean triplet.

• Theorem: If the acute angles of a right-angled triangle have measure 30° and 60°, then the length of the side opposite to 30° angle is half the length of the hypotenuse.
• Theorem: If the acute angles of a right-angled triangle have measure 30° and 60°, then the length of the side opposite to 60° angle is `(sqrt3)/2` × hypotenuse.

• Theorem: If measures of angles of a triangle are 45°, 45°, 90° then the length of each a side containing the right angle is `1/(sqrt2)` × hypotenuse.

• Theorem: In a right angled triangle, if the altitude is drawn to the hypotenuse, then the two triangles formed are similar to the original triangle and to each other.

In a right angled triangle, the perpendicular segment to the hypotenuse from the opposite vertex is the geometric mean of the segments into which the hypotenuse is divided.

In a triangle if the square of one side is equal to the sum of the squares of the remaining two sides, then the triangle is a right angled triangle. 

In Δ ABC, if M is the midpoint of side BC, then AB² + AC² = 2AM² + 2BM²

• Infinite circles pass through one point.
• Infinite circles pass through two distinct points.
• There is a unique circle passing through three non-collinear points.
• No circle can pass through 3 collinear points.

• Tangent theorem: A tangent at any point of a circle is perpendicular to the radius through the point of contact.

Theorem: A line perpendicular to a radius at its point on the circle is a tangent to the circle.

• Theorem: Tangent segments drawn from an external point to a circle are congruent. 

• Theorem of touching circles: If two circles touch each other, their point of contact lies on the line joining their centres.

• Arc of a circle: Major arc and Minor arc
• Central angle
• Measure of an arc

• Two arcs are congruent if their measures and radii are equal.

• Theorem: The chords corresponding to congruent arcs of a circle (or congruent circles) are congruent.
• Theorem: Corresponding arcs of congruent chords of a circle (or congruent circles) are congruent.

• The measure of an inscribed angle is half of the measure of the arc intercepted by it.

• Angles inscribed in the same arc are congruent.
• Angle inscribed in a semicircle is a right angle.

Theorem: Opposite angles of a cyclic quadrilateral are supplementry.

• An exterior angle of a cyclic quadrilateral is congruent to the angle opposite to its adjacent interior angle.

•  If a pair of opposite angles of a quadrilateral is supplementary, the quadrilateral is cyclic.

• Theorem: If two points on a given line subtend equal angles at two distinct points which lie on the same side of the line, then the four points are concyclic.

• Suppose two chords of a circle intersect each other in the interior of the circle, then the product of the lengths of the two segments of one chord is equal to the product of the lengths of the two segments of the
  other chord.

Theorem - The Length of Two Tangent Segments Drawn from a Point Outside the Circle Are Equal

• Division of Line Segment in a Given Ratio
• Construction of a Triangle Similar to a Given Triangle
• To divide a line segment in a given ratio
• To construct a triangle similar to a given triangle as per given scale factor

• To construct a triangle, similar to the given triangle, bearing the given ratio with the sides of the given triangle.
   (i) When vertices are distinct
   (ii) When one vertex is common

1. Using the centre of the circle.
2. Without using the centre of the circle.

• To find distance between any two points on an axis. 
• To find the distance between two points if the segment joining these points is parallel to any axis in the XY plane.

• Division of Line Segment in a Given Ratio
• Construction of a Triangle Similar to a Given Triangle
• To divide a line segment in a given ratio
• To construct a triangle similar to a given triangle as per given scale factor

• Slopes of X-axis, Y-axis and lines parallel to axes.
• Slope of line – using ratio in trigonometry
• Slope of Parallel Lines

• Problems involving Angle of Elevation
• Problems involving Angle of Depression
• Problems involving Angle of Elevation and Depression

• Consideration of the time volume/surface area for solid formed using two or more shapes

• Area of the Sector and Circular Segment
• Length of an Arc