Solution

Q.1 ABC is an Isoceles triangle with AB=AC. A circle through B touching AC at the middle point intersects AB at P. Then AP:AB is:

Solution: First Draw the figure

Now The Informations are AB=AC and D is the middle point of AC so AD=DC=AC/2=AB/2 (Since AB=AC)

Since AD is the tangent to the Circle so 

AD^2= AP*AB

(AC/2)^2 = AP*AB

(AB/2)^2 = AP*AB

AB^2 / 4 = AP*AB

By Dividing both sides by AB^2 we get

1/4 = AP/AB

So the answer is AP:AB= 1:4

Q2. A,B,C and D purchase a gift worth Rs 60. A pays 1/2 of what others are paying, B pays 1/3rd of what others are paying and C pays 1/4th of what others are paying. What is the amount paid by D?

Solution: A+B+C+D=60 -----------> (Equation 1)

Now According to Question A= 1/2 (B+C+D) ------> B+C+D = 2A 

Put the value of B+C+D in Equation 1

A+2A =60 ------> 3A=60-------> A=20

Now again according to Question 

B=1/3 (A+C+D)--------> A+C+D= 3B

By Putting the Value of A+C+D in Equation 1

we get    B+3B= 60------> 4B=60 ---------> B=15

Now again accoring to Question 

C= 1/4 (A+B+D) ---------> A+B+D = 4C

By Putting the Value of A+B+D in equation 1 

We get C+4C = 60------> 5C= 60 ------> C=12

Since A+B+C+D= 60

20+15+12+D=60------> D=60-47=13

Answer : D=13

Solution of (The lengths of three median. Of a triangle are 9cm 12 cm and 15cm.the area of triangle is)

The lengths of three median. Of a triangle are 9cm 12 cm and 15cm.the area of triangle is:

In this triangle medians AD =12, BE =9 and CF=15.

Because the median cut each other on centroid G in the ratio of 2:1 so the segments AG= 8, GD=4, BG=6, GE=3, CG=10 AND GF= 5

Now we take Triangle ABG, in this Triangle we are seeing that one side AG=8 and other side BG= 6 so the third side AB should be 10 and the triangle ABG is a right angled triangle making an angle 90 degree at the point G.

Now take a bigger triangle ABE. In this triangle we know that AG is perpendicular to BE and so the area of the Triangle ABE should be

1/2 AG*BE= 1/2 * 8*9 = 36

Now we know that any median of triangle bisects the triangle into two triangle of equal areas so the median BE bisects the Triangle ABC into two triangles, ABE and BED, of Equal area so 

area of ABE = area of BED = 36 so the area of triangle ABC= area of ABE+ area of BED = 36+36= 72