Test your knowledge with questions about general aspects of Analytical Geometry involving distance between two points, midpoint, straight line equation, among other topics.
Take advantage of the comments in the resolutions to clarify your doubts and gain more knowledge.
question 1
Calculate the distance between two points: A (-2,3) and B (1,-3).
Correct answer: d (A, B) = .
To solve this question, use the formula to calculate the distance between two points.
We substitute the values in the formula and calculate the distance.
The root of 45 is not exact, so it is necessary to carry out rooting until you can no longer remove any number from the root.
Therefore, the distance between points A and B is .
question 2
On the Cartesian plane there are points D (3.2) and C (6.4). Calculate the distance between D and C.
Right answer: .
Being and
, we can apply the Pythagorean Theorem to the DCP triangle.
Substituting the coordinates in the formula, we find the distance between the points as follows:
Therefore, the distance between D and C is
See too: Distance between Two Points
question 3
Determine the perimeter of triangle ABC, whose coordinates are: A (3,3), B (–5, –6) and C (4,–2).
Correct answer: P = 26.99.
1st step: Calculate the distance between points A and B.
2nd step: Calculate the distance between points A and C.
3rd step: Calculate the distance between points B and C.
4th step: Calculate the perimeter of the triangle.
Therefore, the perimeter of triangle ABC is 26.99.
See too: Triangle Perimeter
question 4
Determine the coordinates that locate the midpoint between A (4,3) and B (2,-1).
Correct answer: M (3, 1).
Using the formula to calculate the midpoint, we determine the x coordinate.
The y coordinate is calculated using the same formula.
According to the calculations, the midpoint is (3.1).
question 5
Calculate the coordinates of the vertex C of a triangle, whose points are: A (3, 1), B (–1, 2) and the barycenter G (6, –8).
Correct answer: C (16, –27).
The barycenter G (xGyG) is the point where the three medians of a triangle meet. Its coordinates are given by the formulas:
and
Substituting the x values of the coordinates, we have:
Now we do the same process for y values.
Therefore, vertex C has the coordinates (16,-27).
question 6
Given the coordinates of the collinear points A (-2, y), B (4, 8), and C (1, 7), determine what the value of y is.
Correct answer: y = 6.
For the three points to be aligned, the determinant of the matrix below must be equal to zero.
1st step: replace the x and y values in the matrix.
2nd step: write the elements of the first two columns beside the matrix.
3rd step: multiply the elements of the main diagonals and add them up.
The result will be:
4th step: multiply the elements of the secondary diagonals and invert the sign in front of them.
The result will be:
5th step: join the terms and solve the addition and subtraction operations.
Therefore, for the points to be collinear, the value of y must be 6.
See too: Matrices and Determinants
question 7
Determine the area of triangle ABC, whose vertices are: A (2, 2), B (1, 3) and C (4, 6).
Correct answer: Area = 3.
The area of a triangle can be calculated from the determinant as follows:
1st step: replace the coordinate values in the matrix.
2nd step: write the elements of the first two columns beside the matrix.
3rd step: multiply the elements of the main diagonals and add them up.
The result will be:
4th step: multiply the elements of the secondary diagonals and invert the sign in front of them.
The result will be:
5th step: join the terms and solve the addition and subtraction operations.
6th step: calculate the area of the triangle.
See too: Triangle Area
question 8
(PUC-RJ) Point B = (3, b) is equidistant from points A = (6, 0) and C = (0, 6). Therefore, point B is:
a) (3, 1)
b) (3, 6)
c) (3, 3)
d) (3, 2)
e) (3, 0)
Correct alternative: c) (3, 3).
If points A and C are equidistant from point B, it means that the points are located at the same distance. So, dAB = dCB and the formula to calculate is:
1st step: replace coordinate values.
2nd step: solve the roots and find the value of b.
Hence, point B is (3, 3).
See too: Exercises on distance between two points
question 9
(Unesp) The triangle PQR, in the Cartesian plane, with vertices P = (0, 0), Q = (6, 0) and R = (3, 5), is
a) equilateral.
b) isosceles but not equilateral.
c) scalene.
d) rectangle.
e) obtuse angle.
Correct alternative: b) isosceles but not equilateral.
1st step: calculate the distance between points P and Q.
2nd step: calculate the distance between points P and R.
3rd step: calculate the distance between points Q and R.
4th step: judge the alternatives.
a) WRONG. The equilateral triangle has equal three-sided measurements.
b) CORRECT. The triangle is isosceles, as two sides have the same measurement.
c) WRONG. The scalene triangle has the measurements of three different sides.
d) WRONG. The right triangle has a right angle, that is, 90º.
e) WRONG. The obtuse-angled triangle has one of the angles greater than 90º.
See too: Triangle Classification
question 10
(Unitau) The equation of the straight line that passes through points (3.3) and (6.6) is:
a) y = x.
b) y = 3x.
c) y = 6x.
d) 2y = x.
e) 6y = x.
Correct alternative: a) y = x.
To make it easier to understand, we will call point (3,3) A and point (6,6) B.
Taking P(xPyP) as a point that belongs to the line AB, then A, B and P are collinear and the equation of the line is determined by:
The general equation of the line passing through A and B is ax + by + c = 0.
Substituting the values in the matrix and calculating the determinant, we have:
Therefore, x = y is the equation of the line that passes through the points (3,3) and (6,6).
See too: Line Equation
