Three points

Three points A (-3;-5) B (9;-10) and C (2;k) . AB=AC
What is value of k?

Correct answer:

k₁ =  7
k₂ =  -17

Step-by-step explanation:

$$\begin{gathered} d=\mathrm{\mid }AB\mathrm{\mid } \\ d=\sqrt{(-3-9{)}^2+(-5-(-10){)}^2}=13 \\ {d}^2=(-3-2{)}^2+(-5-k{)}^2 \\ 13^2=(-3-2{)}^2+(-5-k{)}^2 \\ -k^2-10k+119=0 \\ {k}^2+10k-119=0 \\ a=1;b=10;c=-119 \\ D=b^2-4ac=10^2-4\cdot 1\cdot (-119)=576 \\ D>0 \\ {k}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{-10\pm \sqrt{576}}{2}} \\ {k}_{1,2}={\displaystyle \frac{-10\pm 24}{2}} \\ {k}_{1,2}=-5\pm 12 \\ {k}_1=7 \\ {k}_2=-17 \\ \text{ Factored form of the equation: } \\ (k-7)(k+17)=0 \\ {k}_1=7 \end{gathered}$$

Our quadratic equation calculator calculates it.

$k_2=(-17)=-17$

Try another example

Related math problems and questions:

• Vertices of a right triangle
  Show that the points D(2,1), E(4,0), F(5,7) are vertices of a right triangle.
• Distance between 2 points
  Find the distance between the points (7, -9), (-1, -9)
• Calculate 6
  Calculate the distance of a point A[0, 2] from a line passing through points B[9, 5] and C[1, -1].
• Points on line segment
  Points P & Q belong to segment AB. If AB=a, AP = 2PQ = 2QB, find the distance: between point A and the midpoint of the segment QB.
• Distance
  Calculate distance between two points X[18; 19] and W[20; 3].
• On a line
  On a line p : 3 x - 4 y - 3 = 0, determine the point C equidistant from points A[4, 4] and B[7, 1].
• Three points 2
  The three points A(3, 8), B(6, 2) and C(10, 2). The point D is such that the line DA is perpendicular to AB, and DC is parallel to AB. Calculate the coordinates of D.
• Suppose
  Suppose you know that the length of a line segment is 15, x2=6, y2=14 and x1= -3. Find the possible value of y1. Is there more than one possible answer? Why or why not?
• Right triangle from axes
  A line segment has its ends on the coordinate axes and forms with them a triangle of area equal to 36 square units. The segment passes through the point ( 5,2). What is the slope of the line segment?
• Center of line segment
  Calculate the distance of the point X [1,3] from the center of the line segment x = 2-6t, y = 1-4t ; t is .
• Segment
  Calculate the segment AB's length if the coordinates of the end vertices are A[10, -4] and B[5, 5].
• On line
  On line p: x = 4 + t, y = 3 + 2t, t is R, find point C, which has the same distance from points A [1,2] and B [-1,0].
• Triangle IRT
  In isosceles right triangle ABC with right angle at vertex C is coordinates: A (-1, 2); C (-5, -2) Calculate the length of segment AB.
• Line
  Straight-line passing through points A [-3; 22] and B [33; -2]. Determine the total number of points of the line in which both coordinates are positive integers.
• Three points
  Three points K (-3; 2), L (-1; 4), M (3, -4) are given. Find out: (a) whether the triangle KLM is right b) calculate the length of the line to the k side c) write the coordinates of the vector LM d) write the directional form of the KM side e) write the d
• Find the 3
  Find the distance and midpoint between A(1,2) and B(5,5).
• Vertex points
  Given the following points of a triangle: P(-12,6), Q(4,0), R(-8,-6). Graph the triangle. Find the triangle area.