All questions of Where’s the Proof? for Grade 10 Exam

There are two possible solutions for this problem:

1) Solution 1:
To find the distance between two points, we can use the distance formula:

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

In this case, we have two distances:

distance from (-3,0) to (x,y) = sqrt((x - (-3))^2 + (y - 0)^2)
distance from (3,0) to (x,y) = sqrt((x - 3)^2 + (y - 0)^2)

Since the distance from (-3,0) to (x,y) is 4, we can write the equation:

4 = sqrt((x - (-3))^2 + (y - 0)^2)

Simplifying the equation:

16 = (x + 3)^2 + y^2

Similarly, since the distance from (3,0) to (x,y) is 4, we can write the equation:

4 = sqrt((x - 3)^2 + (y - 0)^2)

Simplifying the equation:

16 = (x - 3)^2 + y^2

Now we have a system of equations:

16 = (x + 3)^2 + y^2
16 = (x - 3)^2 + y^2

Expanding the equations:

16 = x^2 + 6x + 9 + y^2
16 = x^2 - 6x + 9 + y^2

Simplifying the equations:

x^2 + 6x + y^2 = 7
x^2 - 6x + y^2 = 7

Subtracting the second equation from the first equation:

0 = 12x

Solving for x:

x = 0

Substituting x = 0 into one of the equations:

0^2 - 6(0) + y^2 = 7

Simplifying the equation:

y^2 = 7

Taking the square root of both sides:

y = ±sqrt(7)

Therefore, the possible values of x and y are:

x = 0, y = sqrt(7) or x = 0, y = -sqrt(7)

So, the correct answer is:

c) x = 0, y = ±sqrt(7)

2) Solution 2:
We can also solve this problem by graphing the two circles with centers (-3,0) and (3,0) and radius 4.

The equation of the first circle is:

(x + 3)^2 + y^2 = 4^2
(x + 3)^2 + y^2 = 16

The equation of the second circle is:

(x - 3)^2 + y^2 = 4^2
(x - 3)^2 + y^2 = 16

By graphing these two circles, we can see that the only point that satisfies both equations is (0, ±sqrt(7)).

Therefore, the correct answer is:

c) x = 0, y = ±sqrt(7)

Quadrant 1 = +,+

Quadrant 2= -,+

Quadrant 3= -,-

Quadrant 4 = +,-

Thus,the point (-1,-5 ) will lie in Quadrant 3 !!!

To find the distance between two points, we can use the distance formula. The distance formula is derived from the Pythagorean theorem and allows us to calculate the distance between two points in a coordinate plane.

The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

In this case, we are given the points A (-6, 7) and B (-1, -5). We need to find the distance 2AB, which means we need to find the distance between A and B and then multiply it by 2.

Let's calculate the distance between A and B first:

x1 = -6, y1 = 7 (coordinates of A)
x2 = -1, y2 = -5 (coordinates of B)

Using the distance formula:
dAB = sqrt((-1 - (-6))^2 + (-5 - 7)^2)
= sqrt(5^2 + (-12)^2)
= sqrt(25 + 144)
= sqrt(169)
= 13

Now, to find the distance 2AB, we multiply the distance between A and B by 2:
2AB = 2 * 13
= 26

So, the distance 2AB is equal to 26. Therefore, the correct answer is option A.

To determine the value of k, we need to check if the three given points (3, 2), (4, k), and (5, 3) are collinear. Two points are collinear if the slope between them is the same as the slope between any other two points on the line.

Calculating the slope between (3, 2) and (4, k):
The slope formula is given by:

m = (y2 - y1) / (x2 - x1)

Using the coordinates (3, 2) and (4, k), we have:

m = (k - 2) / (4 - 3)
m = (k - 2) / 1
m = k - 2

Calculating the slope between (4, k) and (5, 3):
Using the coordinates (4, k) and (5, 3), we have:

m = (3 - k) / (5 - 4)
m = (3 - k) / 1
m = 3 - k

Since the two slopes are equal, we can equate them:

k - 2 = 3 - k

Solving this equation for k:

2k = 5

k = 5/2

Therefore, the value of k is 5/2, which corresponds to option C.

Let's call the two given vertices A and B, with coordinates (3, 2) and (x, y) respectively.

Since a parallelogram has opposite sides that are parallel, we can find the coordinates of the other two vertices by applying the same translation to points A and B.

To find the translation, we can subtract the x-coordinate of point A from the x-coordinate of point B, and subtract the y-coordinate of point A from the y-coordinate of point B:

x - 3 = 3 - 3 = 0
y - 2 = 2 - 2 = 0

So the translation is (0, 0).

To find the coordinates of the other two vertices, we can add the translation to points A and B:

Point A + translation = (3, 2) + (0, 0) = (3, 2)
Point B + translation = (x, y) + (0, 0) = (x, y)

Therefore, the other two vertices of the parallelogram are (3, 2) and (x, y).

Let A(-2, 2)
B(8, -2)
C(-4, -3)
distance
AB=√(x2-x1) ²+(y2-y1) ²
=√(8+2) ²+(-2-2) ²=√100+16=√116
BC=√(-4-8) ²+(-3+2) ²=√144+1=√145
AC=√(-4+2) ²+(-3-2) ²=√4+25=√29
from the above
AB²=(√116) ²=116
BC²=(√145) ²=145
AC²=(√29) ²=29
AB²+AC²=BC²=145units
hence, by Converse of Pythagoras theorem
ABC is a right triangle
and hence,
A, B, C are the vertices of a right ∆
hence, option (C) is correct

Vertices of a Square
The given points (1, 7), (4, 2), (-1, 1), and (-4, 4) can form the vertices of a square.

Properties of a Square
- A square is a special type of rectangle and rhombus.
- All sides of a square are equal in length.
- Opposite sides of a square are parallel.
- All angles in a square are right angles (90 degrees).
- The diagonals of a square are equal in length and bisect each other at right angles.

Verifying the Given Points
To verify if the given points form a square, we need to check the properties mentioned above.
- Calculate the distances between the points to ensure that all sides are equal.
- Check if the slopes of opposite sides are equal to show that they are parallel.
- Calculate the angles between the sides to confirm that they are all right angles.
- Calculate the lengths of the diagonals and check if they are equal.

Conclusion
After verifying the properties of the given points, if all conditions hold true, then the points (1, 7), (4, 2), (-1, 1), and (-4, 4) form the vertices of a square. Hence, the correct answer is option 'D' - square.

B