Segments That Meet at Right Angles

New York State Common Core Math Geometry, Module 4, Lesson 6

Worksheets for Geometry

Student Outcomes

• Students generalize the criterion for perpendicularity of two segments that meet at a point to any two segments in the Cartesian plane.
• Students apply the criterion to determine if two segments are perpendicular.

Segments That Meet at Right Angles

Classwork

Opening Exercise

Carlos thinks that the segment having endpoints 𝐴(0,0) and 𝐵(6,0) is perpendicular to the segment with endpoints 𝐴(0,0) and 𝐶(−2,0). Do you agree? Why or why not?
Working with a partner, given 𝐴(0,0) and 𝐵(3,−2), find the coordinates of a point 𝐶 so that 𝐴𝐶 ⊥ 𝐴𝐵.

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Example

Given points 𝐴(2,2), 𝐵(10,16), 𝐶(−3,1), and 𝐷(4,−3), are 𝐴𝐵 and 𝐶𝐷 perpendicular? Are the lines containing the segments perpendicular? Explain.

Exercises

1. Given 𝐴(𝑎1, 𝑎2), 𝐵(𝑏1, 𝑏2), 𝐶(𝑐1, 𝑐2), and 𝐷(𝑑1, 𝑑2), find a general formula in terms of 𝑎1, 𝑎2, 𝑏1, 𝑏2, 𝑐1, 𝑐2, 𝑑1, and 𝑑2 that will let us determine whether 𝐴𝐵 and 𝐶𝐷 are perpendicular.
2. Recall the Opening Exercise of Lesson 4 in which a robot is traveling along a linear path given by the equation 𝑦 = 3𝑥 − 600. The robot hears a ping from a homing beacon when it reaches the point 𝐹(400,600) and turns to travel along a linear path given by the equation 𝑦 − 600 = −1/3(𝑥 − 400). If the homing beacon lies on the 𝑥-axis, what is its exact location? (Use your own graph paper to visualize the scenario.)
   a. If point 𝐸 is the 𝑦-intercept of the original equation, what are the coordinates of point 𝐸?
   b. What are the endpoints of the original segment of motion?
   c. If the beacon lies on the 𝑥-axis, what is the 𝑦-value of this point, 𝐺?
   d. Translate point 𝐹 to the origin. What are the coordinates of 𝐸′, 𝐹′, and 𝐺′?
   e. Use the formula derived in this lesson to determine the coordinates of point 𝐺
3. A triangle in the coordinate plane has vertices 𝐴(0,10), 𝐵(−8,8), and 𝐶(−3,5). Is it a right triangle? If so, at which vertex is the right angle? (Hint: Plot the points, and draw the triangle on a coordinate plane to help you determine which vertex is the best candidate for the right angle.)
4. 𝐴(−7,1), 𝐵(−1,3), 𝐶(5,−5), and 𝐷(−5,−5) are vertices of a quadrilateral. If 𝐴𝐶 bisects 𝐵𝐷, but 𝐵𝐷 does not bisect 𝐴𝐶, determine whether 𝐴𝐵𝐶𝐷 is a kite.

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