Download presentation

1
**Midpoints: Segment Congruence**

Chapter 1 Section 5 Midpoints: Segment Congruence

2
Warm-Up 1) Name two possible arrangements for G, H, and I on a segment if GH + GI = HI 2) Use the figure below to find each measure. D A C E -10 -8 -6 -4 -2 2 4 6 8 10 a) AC b) DE 3) If M is between L and N, LN = 3x – 1, LM = 4, and MN = x – 1, find MN. 4) What is the length of ST for S(-1, -1) and T(4, 6)?

3
**2) Use the figure below to find each measure.**

1) Name two possible arrangements for G, H, and I on a segment if GH + GI = HI H, G, I or I, G, H 2) Use the figure below to find each measure. D A C E -10 -8 -6 -4 -2 2 4 6 8 10 a) AC A= 1, C = 5 A – C 1 – 5 = -4 So AC is 4. b) DE D = -1, E = 8 D – E -1 – 8 = -9 So DE is 9.

4
**Use the segment addition Postulate. LM + MN = LN 4 + x -1 = 3x - 1 **

3) If M is between L and N, LN = 3x – 1, LM = 4, and MN = x – 1, find MN. Use the segment addition Postulate. LM + MN = LN 4 + x -1 = 3x - 1 x + 3 = 3x - 1 3 = 2x - 1 4 = 2x 2 = x Now plug 2 in for x in the equation for MN MN = x - 1 MN = 2 - 1 MN = 1 L M N

5
**4) What is the length of ST for S(-1, -1) and T(4, 6)?**

Distance Formula d=√((x2 – x1)2 + (y2 – y1)2) Pick one point to be x1 and y1 and the other point will be x2 and y2. Let point S be x1 and y1 and point T be x2 and y2. d=√((4 – -1)2 + (6 – -1)2) d=√((4 + 1)2 + (6 + 1)2) d=√((5)2 + (7)2) d= √((25) + (49)) d= √(74) So the distance between the two points is √(74) or about 8.6.

6
Vocabulary P M Q Midpoint- The midpoint M of PQ is the point between P and Q such that PM = MQ Segment bisector- Any segment, line, or plane that intersects a segment at its midpoint. Line L is a segment bisector. Theorems- A statement that must be proven. Proof- A logical argument in which each statement you make is backed up by a statement that is accepted as true. L

7
Vocabulary Cont. Midpoint Formulas- On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is (a + b)/2. In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have the coordinates (x1, y1) and (x2, y2) are [(x1 + x2)/2, (y1 + y2)/2]. Midpoint Theorem- If M is the midpoint of line AB, then Segment AM congruent to segment MB. A M B

8
Example 1: If the coordinate of H is -5 and the coordinate of J is 4, what is the coordinate of the midpoint of line HJ? H and J are on a number line so use the equation (a + b)/2. Let point H be a and point J be b. (a + b)/2 (-5 + 4)/2 -1/2 So the coordinate of the midpoint is at -1/2.

9
Example 2: If the coordinate of H is -10 and the coordinate of J is 2, what is the coordinate of the midpoint of line HJ? H and J are on a number line so use the equation (a + b)/2. Let point H be a and point J be b. (a + b)/2 ( )/2 -8/2 -4 So the coordinate of the midpoint is at -4.

10
**Example 3: Find the coordinates of the midpoint of line VW for V(3, -6) and W(7, 2).**

V and W are on a coordinate plane so use the equation [(x1 + x2)/2, (y1 + y2)/2]. Let point V be x1 and y1 and let point W be x2 and y2. (x1 + x2)/2 = x-coordinate of the midpoint (3 + 7)/2 10/2 5 (y1 + y2)/2 = y-coordinate of the midpoint (-6 + 2)/2 (-4)/2 -2 So the midpoint of line VW is at the point (5,-2)

11
**Example 4: Find the coordinates of the midpoint of line VW for V(4, -2) and W(8, 6).**

V and W are on a coordinate plane so use the equation [(x1 + x2)/2, (y1 + y2)/2]. Let point V be x1 and y1 and let point W be x2 and y2. (x1 + x2)/2 = x-coordinate of the midpoint (4 + 8)/2 12/2 6 (y1 + y2)/2 = y-coordinate of the midpoint (-2 + 6)/2 (4)/2 2 So the midpoint of line VW is at the point (6,2)

12
**Example 5: The midpoint of line RQ is P(4, -1)**

Example 5: The midpoint of line RQ is P(4, -1). What are the coordinates of R if Q is at (3, -2)? R and Q are on a coordinate plane so use the equation [(x1 + x2)/2, (y1 + y2)/2]. Let point R be x1 and y1 and let point Q be x2 and y2. (x1 + x2)/2 = x-coordinate of the midpoint (x1 + 3)/2 = 4 x1 + 3 = 8 x1 = 5 (y1 + y2)/2 = y-coordinate of the midpoint (y1 + -2)/2 = -1 (y1 + -2) = -2 y1 = 0 So point R is at (5,0).

13
**Example 6: The midpoint of line RQ is P(4, -6)**

Example 6: The midpoint of line RQ is P(4, -6). What are the coordinates of R if Q is at (8, -9)? R and Q are on a coordinate plane so use the equation [(x1 + x2)/2, (y1 + y2)/2]. Let point R be x1 and y1 and let point Q be x2 and y2. (x1 + x2)/2 = x-coordinate of the midpoint (x1 + 8)/2 = 4 x1 + 8 = 8 x1 = 0 (y1 + y2)/2 = y-coordinate of the midpoint (y1 + -9)/2 = -6 (y1 + -9) = -12 y1 = -3 So point R is at (0,-3).

14
**Plug 3 in for x in the equation for XY. XY = 16x – 6 XY = 16(3) – 6 **

Example 7: U is the midpoint of line XY. If XY = 16x – 6 and UY = 4x + 9, find the value of x and the measure of line XY. X U Y Since U is the midpoint of line XY, we can use the midpoint formula. The midpoint formula tells us that XU is congruent or equal to UY. So XU + UY = XY; UY + UY = XY or 2(UY) = XY. 2( UY) = XY 2(4x + 9) = 16x – 6 8x + 18 = 16x – 6 18 = 8x – 6 24 = 8x 3 = x Plug 3 in for x in the equation for XY. XY = 16x – 6 XY = 16(3) – 6 XY = 48 – 6 XY = 42

15
**Plug 4 in for x in the equation for XY. XY = 2x + 14 XY = 2(4) + 14 **

Example 8: U is the midpoint of line XY. If XY = 2x + 14 and UY = 4x - 5, find the value of x and the measure of line XY. X U Y Since U is the midpoint of line XY, we can use the midpoint formula. The midpoint formula tells us that XU is congruent or equal to UY. So XU + UY = XY; UY + UY = XY or 2(UY) = XY. 2( UY) = XY 2(4x - 5) = 2x + 14 8x - 10 = 2x + 14 6x - 10 = 14 6x = 24 4 = x Plug 4 in for x in the equation for XY. XY = 2x + 14 XY = 2(4) + 14 XY = XY = 22

16
**Plug 8 in for x in either of the equations. XY = 2x + 11 **

Example 9: Y is the midpoint of line XZ. If XY = 2x + 11 and YZ = 4x - 5, find the value of x and the measure of line XZ. X Y Z Since Y is the midpoint of line XZ, we can use the midpoint formula. The midpoint formula tells us that XY is congruent or equal to YZ. So XY + YZ = XZ; XY + XY = XZ or 2(XY) = XZ. XY = YZ 2x + 11 = 4x - 5 11 = 2x - 5 16 = 2x 8 = x Plug 8 in for x in either of the equations. XY = 2x + 11 XY = 2(8) + 11 XY = XY = 27 2(XY) = XZ 2(27) = XZ 54 = XZ

17
**Plug 2 in for x in either of the equations. XY = -3x + 9 **

Example 9: Y is the midpoint of line XZ. If XY = -3x + 9 and YZ = 4x - 5, find the value of x and the measure of line XZ. X Y Z Since Y is the midpoint of line XZ, we can use the midpoint formula. The midpoint formula tells us that XY is congruent or equal to YZ. So XY + YZ = XZ; XY + XY = XZ or 2(XY) = XZ. XY = YZ -3x + 9 = 4x - 5 9 = 7x - 5 14 = 7x 2 = x Plug 2 in for x in either of the equations. XY = -3x + 9 XY = -3(2) + 9 XY = XY = 3 2(XY) = XZ 2(3) = XZ 6 = XZ

Similar presentations

Presentation is loading. Please wait....

OK

Distance and Midpoints

Distance and Midpoints

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on market value added Ppt on natural resources and conservation job Free download ppt on statistics for class 11 Ppt on chapter 3 drainage pipe Ppt on the parliament of india Solar system for kids ppt on batteries Ppt on economic development in india 2012 Ppt on electronic media Ppt on sources of energy for class 8th maths Mba ppt on international monetary system