Midpoints: Segment Congruence Chapter 1 Section 5 Midpoints: Segment Congruence

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)?

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.

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

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.

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

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

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.

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 (-10 + 2)/2 -8/2 -4 So the coordinate of the midpoint is at -4.

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)

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)

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).

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).

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

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 = 8 + 14 XY = 22

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 = 16 + 11 XY = 27 2(XY) = XZ 2(27) = XZ 54 = XZ

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 = -6 + 9 XY = 3 2(XY) = XZ 2(3) = XZ 6 = XZ