Free Coordinate Geometry 02 Practice Test - 10th Grade
Question 1
The points (a,a),(–a,−a) and (–√3a,√3a) are the vertices of a/an _____ triangle.
scalene
right angled
isosceles right angled
equilateral
SOLUTION
Solution : D
Let's name the points as A(a,a), B(−a,−a) and C(–√3a,√3a).
By distance formula, distance between points (x1,y1) and (x2,y2)
=√(x2−x1)2+(y2−y1)2
BC=√(−√3a+a)2+(√3a+a)2 =√a2+(√3a)2−2×a×√3a+√3a)2+a2+2×a×√3aBC=√8a2=2√2a units Similarly AB=√(a+a)2+(a+a)2 =√8a2=2√2a units AC=√(−√3a−a)2+(√3a−a)2 =√8a2=2√2a unitsWe get, AB=BC=AC=2√2a units ∴△ABC is an equilateral triangle.
Question 2
The point C (–1,2) divides the line segment AB in the ratio 3:4, where the coordinates of A is (2, 5). The coordinates of B are___________.
(–5, –2)
(–1, – 2)
(–4, – 2)
(5, – 4)
SOLUTION
Solution : A
If a point P(x,y) divides a line segment joining (x1,y1) and (x2,y2) in the ratio m:n, then the coordinates of P are given by:
x=mx2+nx1m+n, y=my2+ny1m+n
Let C(−1,2) divide the line joining A(2,5) and B(x,y) in the ratio 3:4.
Then,
C(3x+87,3y+207) = C(-1, 2)
⇒3x+87=−13x+8=−7⇒x=−5 ⇒3y+207=23y+20=14⇒y=−2
Thus, the coordinates of B are B(−5,−2).
Question 3
Midpoint of the line joining the points (x1,y1) and (x2,y2) is (x1+x22,y1+y22).
True
False
SOLUTION
Solution : A
True. Midpoint of the line joining the points (x1,y1) and (x2,y2) is (1(x1)+1(x2)1+1,1(y1)+1(y2)1+1) = (x1+x22,y1+y22).
Question 4
M (-2, y) is a point equidistant from the coordinates A(5,7) and B(3,-4). Find the value of y.
SOLUTION
Solution : A
Point M (-2, y) is equidistant from the points A (5, 7) and B(3 , -4).
Distance AM = Distance BM
⇒√((xa−xm)2+(ya−ym)2)) = √((xb−xm)2+(yb−ym)2))
⇒(5−(−2))2 + (7−y)2 = (3−(−2))2 + (−4−y)2
⇒49+49−14y+y2=25+16+8y+y2
⇒22y=57
⇒y=5722
Question 5
Find the area of the triangle with coordinates A(-2, 1), B(3, 3) and C(1, -2).
10 sq. units
10.5 sq. units
11 sq. units
11.5 sq. units
SOLUTION
Solution : B
Area of a triangle having vertices (x1,y1),(x2,y2) and (x3,y3)
=12[x1(y2−y3)+x2(y3−y1)+x3(y1−(y2)]
Given, (x1,y1) = (-2, 1)(x2,y2) = (3, 3)
(x3,y3) = (1, -2)
=12[−2(3+2)+3(−2−1)+1(1−3)]=−10.5Since area is not a negative quantity, take the absolute value.
Thus, area of the given triangle = 10.5 sq. units.
Question 6
A(a,b), B(c,d) and C(e,f) are the vertices of a triangle.
i) AB + BC > AC
ii) Area of the triangle = (12)[a(c-e)+c(f-b)+e(b-d)]
Which of the following are true?
Only (i) is true
Only (ii) is true
Both (i) and (ii) is true
Neither (i) nor (ii) is true
SOLUTION
Solution : A
Since A, B and C are the vertices of a triangle, it follows that sum of two sides is greater than the third side. So (i) is correct.
And since the area of a triangle with vertices (x1, y1), (x2 , y2 ) and (x3 , y3 ) is
12[ x1(y2 - y3) + x2( y3 - y1) + x3(y1 - y2)]
= 12 [a (d - f) + c (f - b) + e (b - d)] (by putting x1 = a, y1 = b, x2 = c, y2 = d, x3 = e, y3 = f)
Since the above area is not the same as that given in (ii), (ii) is incorrect.
Question 7
The coordinate of the mid-point of the line segment joining the points (2,1) and (1,-3) is ____.
(32,−1)
(−3,−2)
(3,−2)
(4,−1)
SOLUTION
Solution : A
Let, (x,y) be coordinates of the mid-point.
From the section formula if (x,y) are the midpoints of the line segment joining (x1,y1) and (x2,y2),then x=x1+x22 and y=y1+y22
x=2+12 and y=−3+12
So, (x,y)=(32,−1)
Question 8
The distance between the points (-1, 5) and (2, 1) is
SOLUTION
Solution :Distance formula for the points (x1,y1)and(x2,y2) = √(x2−x1)2+(y2−y1)2
∴ Distance between the points (-1, 5) and (2, 1),
d = √(2−(−1))2+(1−5)2
= √(2+1)2+(−4)2
= √9+16 = √25
= 5 units
Question 9
The point (2, 0) lies in both the 1st and 4th quadrants.
SOLUTION
Solution : B
False. Any point that lies on the coordinate axes doesn't belong to any of the quadrants. The point (2,0) lies on the coordinate axes.
Question 10
The area of a triangle is 5 square units. Two of its vertices are (2, 1) and (3, -2) and the third vertex lies on y = x + 3, the third vertex is
(3,4)
(52, 132)
(72, 132)
SOLUTION
Solution : C and D
Given, area of triangle = 5 sq. units.
Let the third vertex be (x,y).
Area of a triangle formed by (x1,y1), (x2,y2) & (x3,y3)=12|x1(y2−y3)+x2(y3−y1)+x3(y1−y2)|5=12|2(−2−y)+3(y−1)+3x|5=12|−4−2y+3y−3+3x|
3x+y−7=10 or 3x+y−7=−10⇒3x+y=17 or 3x+y=−3
Also given that the third vertex lies on y=x+3. It means that the point of intersection of both the lines is the vertex.
Let us now solve y=x+3⇒x−y=−3 −−−(1) and 3x+y=17 −−−(2).
Multiplying (1) by 3, ⇒3x−3y=−9 −−−(3)
(2)−(3)⇒ 3x+y= 17−3x∓3y=∓9––––––––––––––––– 4y=26 y=132.
Substituting this value of y in (1).
x=y−3=132+3=72
∴x=72 and y=132
Now, to solve 3x−3y=−9−−−(3) and 3x+y=−3−−−(4)
(4)−(3)⇒ 3x+y =−3−3x∓3y=∓9––––––––––––––––– 4y=6 y=32.
Substituting this value of y in (1).
x=y−3=32−3=−32
∴x=−32 and y=32.
Vertex are (72,132) & (−32,32)