P and Q are the mid-point of the sides CA and CB respectively of a $\Delta ABC$, right angled at C. Prove that:

(i) $4AQ^2 = 4AC^2 + BC^2$

(ii)$4BP^2 = 4BC^2 + AC^2$

(iii) $4(AQ^2 + BP^2) = 5AB^2$.

Solution

The areas of two similar triangles are in the ratio of the squares of the corresponding altitude

The areas of two similar triangles are in the ratio of the squares of the corresponding median.

The areas of two similar triangles are in the ratio of the squares of the corresponding angle bisector segments.

If, AD BE and CF are medians of the $ \Delta ABC$, then prove that

$3 (AB^2 + BC^2 + CA^2) = 4(AD^2 + BE^2 + CF^2)$

Solution

In an equilateral triangle ABC, D is a point on side BC such that $BD = \frac {1}{3} BC$. Prove that $9AD^2 = 7AB^2$

Solution

O is a point in the interior of a triangle ABC, $OD \perp BC$, $OE \perp AC$ and $OF \perp AB$. Show that:

a. $OA^2 + OB^2 + OC^2 - OD^2 - OE^2 - OF^2= AF^2 + BD^2 + CE^2$.

b. $AF^2 + BD^2 + CE^2 = AE^2 + CD^2 + BF^2$

Solution

D is a point on the side BC of a triangle ABC such that $ \angle ADC = \angle BAC$. Show that $CA^2 = CB.CD$

Solution

D, E and F are respectively the mid-points of sides AB, BC and CA of $ \Delta ABC$. Find the ratio of the areas of $ \Delta DEF$ and $ \Delta ABC$

Solution

E is a point on side CB produced of an isosceles triangle ABC with AB = AC. If $AD \perp BC$ and $EF \perp AC$,

prove that $\Delta ABD \sim \Delta ECF$

Solution

O is any point inside a rectangle ABCD. Prove that $OB^2 + OD^2= OA^2 + OC^2$.

Solution

ABCD is a rectangle. Points M and N are on BD such that $AM \perp BD$ and $CN \perp BD$. Prove that $BM^2 + BN^2 = DM^2 + DN^2$

Solution

- Similar Figures
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- Similar Polygons
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- Basic Proportionally Theorem (or Thales Theorem)
- |
- Criteria for Similarity of Triangles
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- Different Criterion for similarity of the triangles
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- Areas of Similar Triangles
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- Pythagoras Theorem
- |
- Converse of Pythagoras Theorem

Given below are the links of some of the reference books for class 10 math.

- Oswaal CBSE Question Bank Class 10 Hindi B, English Communication Science, Social Science & Maths (Set of 5 Books)
- Mathematics for Class 10 by R D Sharma
- Pearson IIT Foundation Maths Class 10
- Secondary School Mathematics for Class 10
- Xam Idea Complete Course Mathematics Class 10

You can use above books for extra knowledge and practicing different questions.

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