# Segments That Meet at Right Angles

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

Worksheets for Geometry, Module 4, Lesson 6

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 π΄(0,0) and π΅(6,0) is perpendicular to the segment with endpoints π΄(0,0) and πΆ(β2,0). Do you agree? Why or why not?
Working with a partner, given π΄(0,0) and π΅(3,β2), find the coordinates of a point πΆ so that π΄πΆ β₯ π΄π΅.

Example

Given points π΄(2,2), π΅(10,16), πΆ(β3,1), and π·(4,β3), are π΄π΅ and πΆπ· perpendicular? Are the lines containing the segments perpendicular? Explain.

Exercises

1. Given π΄(π1, π2), π΅(π1, π2), πΆ(π1, π2), and π·(π1, π2), find a general formula in terms of π1, π2, π1, π2, π1, π2, π1, and π2 that will let us determine whether π΄π΅ and πΆπ· are perpendicular.
2. Recall the Opening Exercise of Lesson 4 in which a robot is traveling along a linear path given by the equation π¦ = 3π₯ β 600. The robot hears a ping from a homing beacon when it reaches the point πΉ(400,600) and turns to travel along a linear path given by the equation π¦ β 600 = β1/3(π₯ β 400). If the homing beacon lies on the π₯-axis, what is its exact location? (Use your own graph paper to visualize the scenario.)
a. If point πΈ is the π¦-intercept of the original equation, what are the coordinates of point πΈ?
b. What are the endpoints of the original segment of motion?
c. If the beacon lies on the π₯-axis, what is the π¦-value of this point, πΊ?
d. Translate point πΉ to the origin. What are the coordinates of πΈβ², πΉβ², and πΊβ²?
e. Use the formula derived in this lesson to determine the coordinates of point πΊ
3. A triangle in the coordinate plane has vertices π΄(0,10), π΅(β8,8), and πΆ(β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. π΄(β7,1), π΅(β1,3), πΆ(5,β5), and π·(β5,β5) are vertices of a quadrilateral. If π΄πΆ bisects π΅π·, but π΅π· does not bisect π΄πΆ, determine whether π΄π΅πΆπ· is a kite.

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