Question 10-12 with answers

10.
In the xy-coordinate system, if (a,b) and (a+3, b+k) are two points on the line defined by the equation x = 3y-7, then
k=  (A) 9  (B) 3  (C) 7/3  (D) 1  (E) 1/3

Solution:
x=3y-7 
Re-write the above equation to slope intercept form (y=mx+b)
where m is the slope. 
(x+7)/3=y 
m=1/3 
m=y2-y1/x2-x1 
1/3=k/3 
1=k

11.
Peter and Paul start simultaneously on 2 different cars from Point A and travel towards Point B at speeds of 52 kmph and 39 kmph respectively on the same road. As soon as Peter reaches Point B, he returns back to Point A on the same road and meets Paul on the way. How far from Point B do the two friends meet, if the distance between the 2 points is 70 kms?

Solution:

Since they are meeting after sometime say T. Therefore, compare the time taken by both peter and paul.  70+x is covered by peter and 70-x is covered by paul. 
(70+x)/52 = (70-x)/39
 this gives x=10.

12.
In triangle ABC, AD is the bisector of |A, AB=10 cm, AC=14 cm and area of triangle ABD = 140 sq cm. Find area of triangle ACD

Solution:

An angle bisector of any angle between 2 sides of a triangle divides the Area of the triangle into the ratio of sides . 
Area of any triangle is 1/2 *(Product of any 2 sides of the triangle) * (Sin of Angle between those 2 sides)  
Now coming to the question at concern. 
Here area of ABD => 140 = 1/2*(AB * AD) *(Sin of angle BAD) ---eqn (1) 
Area of ACD = 1/2*(AC*AD) * (Sin of angle DAC) ---eqn(2) 
angle DAC = angle BAD ---eqn(3)
as angle A is bisected 

Using eqn 1 and 2 and 3, gives 196 as area of ACD.