Solved Trigonometry Problems: 10th Grade

Here is a list of few good problems at the 10th grade. Get the full solution below and if you want
more comment or mail us at maths.programming@gmail.com!


Download here

Geometry Inequality

Post by Maths.

Prove that in any quadrilateral, the sum of the diagonals is greater than the half of its perimeter.

Consider the quad. In the above diagram. Let E be the point of the intersection.

Now, AE+EB > AB

EB+EC > BC

AE+ED > AD

EC+ED > DC (Using Triangle Inequality)

Adding the above four inequalities we get

2(AE+EC+EB+ED) > AB+BC+AD+DC

=> AC + BD > ½(AB+BC+AD+DC)

Thus sum of the diagonals is greater than the half of its perimeter Q.E.D

In any triangle four times the sum of its medians is greater than 3 times its perimeter.

We know that difference of any two sides of a triangle is less than the third side (prove it)

In triangle ABE,

AE > AB-BE

In triangle ACE,

AE > AC-CE

Adding above two inequalities we get,

2AE > AB + AC -(BE+CE)

=> AE > ½(AB+AC-BC)

=> 4AE > 2(AB+AC-BC).........(1)

Similarly,

4BD > 2(AB+BC-AC).............(2) and 4CF > 2(AC+BC-AB)........(3)

Adding (1),(2) and (3) we have,

4(AE+BD+CF) > 2(AB+AC-BC+AB+BC-AC+AC+BC-AB)

=> 4(AE+BD+CF) > 2(AC+AB+BC)

=> sum of the lengths of the medians is greater than half the perimeter

We can strengthen the inequality by using the fact that the point 'O' divides the medians AE,BD,CF internally in the ration 2:1

Therefore, OD:OB = 1:2

=> (OB+OD):OB = (1+2):2

=> BD:OB=3:2

=> OB = 2/3 BD........(a)

Similarly, OC = 2/3 CF.........(b) and OA = 2/3 AE.......(c)

Now in triangle OBC, OB+OC> BC

=>2/3(BD+CF)>BC [using (a) and (b)]

=> 2(BD+CF)>3BC

Similarly, 2(CF+AE)>3AC and 2(BD+AE)>3AB

Adding the last three inequalities we get 4(AE+BD+CF) > 3(AB+BC+CA)

In the triangle ABC, AE,BD and CF are the medians where O is the point of there intersection

Solved Problems on Circles (Tangent Properties)

In these material we explore the properties of circles, tangents and common tangents. Using congruency and similarity of triangles many of the desired property has been deduced.

Post by Maths.

Coordinate Geometry

Solved Problems on Coordinate Geometry for 10th grade. Problems based on distance formula, section ratio formula   

 

GEOMETRY Thales Theorem or Basic Proportionality Theorem Solved Problems

Solved Problems on Thales Theorem useful for students of 10th grade of CBSE and ICSE and other state board. This material is also useful for the preparation of Regional Mathematics Olympiad. Basic Geometry is a pre-requisite  for any Olympiad.

Probability Solved problems for CBSE, ICSE 10th grade

Post by Maths.