# Lesson 12: Dividing Segments Proportionately

## Transcription

Lesson 12: Dividing Segments Proportionately

GEOMETR Lesson 12 NYS COMMON CORE MATHEMATSCS CURRICULUM Period: Date Lesson 12: Dividing Segments Proportionately Learning Target • I can find midpoints of segments and points that divide segments two or more proportional or equal parts. Opening Exercise (15 minutes) Points/I(-4,5), S(12,13) and C(12,9) are plotted on the coordinate grid • What is the length of ~AC1 1(0 • What is the length of BC? I 1 2 1 4 S I T t * II 11 II II Mark the halfway point on AC and label it point . What are the coordinates of pointP? Mark the halfway point on BC and label it point R. What are the coordinates of point R~> Draw a segment from P to AB perpendicular to AC. Mark the intersection point M. What are the coordinates of M? Draw a segment from R to >15 perpendicular to BC. Mark the intersection point M. What are th< coordinates of M? Point M is called the of AB. Look at the coordinates of the endpoints and the midpoint. Can you describe how to find the coordinates of the midpoint knowing the endpoints algebraically? The general formula for the midpoint of a segment with endpoints i,yi) and (x2,y2) using the average formula: M (——-,——-J (*,.»,) (x2,y,) GEOMETR Lesson 12 NYS COMMON CORE MATHEMATICS CURRICULUM Period: Name Date Example 1. Find the midpoint of ST given 5(-2,8) and T(10, -4). (Sketch the situation) K "<• T M\(' ) / — Example 2. Find the point that is one-quarter of the way along of^ST given 7(10, -4). (Sketch the situation) 10 Example 3. Af (— 2, 10) is the midpoint of segment AB. If A has coordinates (4, —5), what are the coordinates of B2~ ~v ~T~O ~~ y- *-ir y Example 4 . Given PQ and point R that lies on PQ such that point R lies - of the length of PQ from point P along PQ. -? Use the given information to determine the following ratios: PR-.PQ = ' RQ:PQ= 2^/Q PR-.RQ = RQ:PR = GEOMETR NYS COMMON CORE MATHEMATICS CURRICULUM LSSSOH 12 Period: Name Date Examples. Given points A (—3,5) and 5(12,15), find the coordinates of the point, C, that sits -of the -way along the segment AS, closer to A than it is to B. (Sketch the situation) Y 10 >/ Divide the segment based on a part:whole ratio To partition (divide) the segment into smaller parts when part to whole ratio is given: 1. Multiple the difference in x-coordinates (x2 — *i) and the difference in y-coordi nates (y2 — yj.) by / part . whole the given ratio ( - ). Then add those products to the original point (*i,yi) to find the partition point (xp,yp). Another way to look at it: Recall point-slope form of a linear equation, y — y± = m(x — x-^). By adding to both sides, you get = m(x - xj + y^ . Finding a partition point uses a similar formula: and Division of the segment given as a part: part ratio -» convert to part:whole Example 6. Find the point on the directed segment from (—3,0) to (5,8) that divides it in the ratio of 1: 3. (Sketch the situation) Example 7. Find the point on the directed segment from (-4,5) to (12,13) that divides it into a ratio of 1:7. (Sketch the situation) \ / \X D&/T v » i I'") -*^~^ » ~%Z i ~\— U \i~ l> A > N * *" ^- ' *v ^/¥ GEOMETR NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 12 Name Period: Date CB 1 Example 8. Given points ,4(3, —9) and £(19, — 1), find the coordinates of point C such that — = -. AC 9^ (Sketch the situation) *- -7 -? 7 -7 Example 9. Find the coordinates of point P along the directed line 3 segment AB so that the ratio of AP to PB is -. {h'T~ J, 1,1ft- _: •I V ? o CB 3 Example 10. Given points 4(3, -5) and fl(19, -1), find the coordinates of point C such that — = -. 4C etch the situation V £ 7 ^