Math Problem Statement

Given points A(2, 3), B(k, 1), and C(-1, 0), find the value of k such that angle ∠B is 90°, i.e., AB ⊥ BC.

Solution

The problem in the image is asking to find the value of kk for which B^=90\angle \hat{B} = 90^\circ, meaning that the triangle has a right angle at point BB.

The points of the triangle are:

  • A(2,3)A(2, 3)
  • B(k,1)B(k, 1)
  • C(1,0)C(-1, 0)

To solve this, we need to use the concept that two lines are perpendicular if the product of their slopes is equal to 1-1.

Steps:

  1. Find the slope of line ABAB: The slope of a line between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by: mAB=y2y1x2x1m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} For A(2,3)A(2, 3) and B(k,1)B(k, 1), the slope of ABAB is: mAB=13k2=2k2m_{AB} = \frac{1 - 3}{k - 2} = \frac{-2}{k - 2}

  2. Find the slope of line BCBC: Using the points B(k,1)B(k, 1) and C(1,0)C(-1, 0), the slope of BCBC is: mBC=10k(1)=1k+1m_{BC} = \frac{1 - 0}{k - (-1)} = \frac{1}{k + 1}

  3. Set up the perpendicularity condition: Since B^=90\angle \hat{B} = 90^\circ, the lines ABAB and BCBC must be perpendicular, meaning: mABmBC=1m_{AB} \cdot m_{BC} = -1 Substituting the slopes: 2k21k+1=1\frac{-2}{k - 2} \cdot \frac{1}{k + 1} = -1

  4. Solve for kk: Now solve the equation: 2(k2)(k+1)=1\frac{-2}{(k - 2)(k + 1)} = -1 Multiply both sides by (k2)(k+1)(k - 2)(k + 1): 2=(k2)(k+1)-2 = -(k - 2)(k + 1) Simplify the right-hand side: 2=(k2k2)-2 = -(k^2 - k - 2) Distribute the negative sign: 2=k2+k+2-2 = -k^2 + k + 2 Bring all terms to one side: k2k4=0k^2 - k - 4 = 0 Now, solve this quadratic equation using the quadratic formula: k=(1)±(1)24(1)(4)2(1)=1±1+162=1±172k = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-4)}}{2(1)} = \frac{1 \pm \sqrt{1 + 16}}{2} = \frac{1 \pm \sqrt{17}}{2} So, the two possible values for kk are: k=1+172ork=1172k = \frac{1 + \sqrt{17}}{2} \quad \text{or} \quad k = \frac{1 - \sqrt{17}}{2}

Would you like more details on the steps or another part of the solution explained?

Here are some related questions for further practice:

  1. What is the equation of a line passing through points A(2,3)A(2, 3) and C(1,0)C(-1, 0)?
  2. How can we verify that the triangle is right-angled using the distance formula?
  3. What is the area of the triangle formed by points A(2,3)A(2, 3), B(k,1)B(k, 1), and C(1,0)C(-1, 0)?
  4. How does the value of kk affect the orientation of the triangle?
  5. If A^=90\angle \hat{A} = 90^\circ, how would we find the value of kk?

Tip: Always check the perpendicularity condition by verifying the product of the slopes when solving geometry problems involving right angles.

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Math Problem Analysis

Mathematical Concepts

Geometry
Coordinate Geometry
Perpendicular Slopes

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Perpendicular lines condition: m1 * m2 = -1
Quadratic equation: ax^2 + bx + c = 0

Theorems

Perpendicular slope theorem
Quadratic formula

Suitable Grade Level

Grades 9-10