PREAP BELLWORK BELLWORK SHEET 7 1 Define midsegment

PRE-AP BELLWORK (BELLWORK SHEET # 7) 1) Define midsegment of a triangle. Draw a picture to illustrate a midsegment. (use a ruler to draw)

RELATIONSHIPS WITHIN TRIANGLES 5 -1 MIDSEGMENTS OF TRIANGLES Coach Patterson

MIDSEGMENTS OF TRIANGLES Midsegment of a triangle �A segment connecting the midpoints of two sides of the triangle. � Every triangle has three segments, which form the midsegment triangle.

MIDSEGMENTS OF TRIANGLES EX. Lets examine midsegments in the coordinate plane. The vertices of ΔXYZ are X(– 1, 8), Y(9, 2), and Z(3, – 4). M and N are the midpoints of XZ and YZ. Show that MN ││ XY and MN = ½ XY. Step 1: Find the coordinates of M and N. Step 2: Compare the slopes of MN and XY. Step 3: Compare the heights of MN and XY.

MIDSEGMENTS OF TRIANGLES Step 1: Find the coordinates of M and N. � Use the Midpoint formula.

MIDSEGMENTS OF TRIANGLES Step 2: Compare the slopes of MN and XY. Since the slopes of the two lines are the same, the lines are parallel to each other.

MIDSEGMENTS OF TRIANGLES Step 3: Compare the heights of MN and XY. � Use � XY the distance formula. is double the size of MN, or MN is half the size of XY.

MIDSEGMENTS OF TRIANGLES OYO… The vertices of ΔRST are R(– 7, 0), S(– 3, 6), and T(9, 2). M is the midpoint of RT, and N is the midpoint of ST. Show that MN ││ RS and MN = ½ RS. Step 1: Find the coordinates of M and N. Step 2: Compare the slopes of MN and XY. Step 3: Compare the heights of MN and XY. Use a partner and complete the exercises as we did in the previous example.

MIDSEGMENTS OF TRIANGLES Triangle Midsegment Theorem

MIDSEGMENTS OF TRIANGLES Find the length of PQ. Since P and Q are both midpoints, then PQ is parallel to BC and half its length. Therefore, PQ is 3.

MIDSEGMENTS IN TRIANGLES Find the x. Since B and C are midpoints, then BC is parallel to DE and half the length of DE or DE is twice the length of BC. x – 1 = 12 x = 13

MIDSEGMENTS IN TRIANGLES If AB is 40 cm, find the length of XY. � 80 cm. Can we use the Triangle Midsegment Theorem to find the lengths of ZX and ZY? � No. We are unsure if C is the midpoint of XY.

MIDSEGMENTS IN TRIANGLES Assume X is the midpoint of RT and Y is the midpoint of TS. Find x. Since XY is half the length of RS, then 2 x – 6 = 9 2 x = 15 x = 7. 5

BISECTORS IN TRIANGLES Triangles are important in the relationships involving perpendicular bisectors and angle bisectors. Perpendicular Bisector Theorem � If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. � Also, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisectors of the segment.

BISECTORS IN TRIANGLES

BISECTORS IN TRIANGLES Assume XW is congruent to WZ. How is YW related to XZ. � YW is the perpendicular bisector of XZ. What is the length of WZ? 3 x + 25 � 7 Find x. Find YX. Find YZ. 7 4 x – 12

BISECTORS IN TRIANGLES To find x: 3 x + 25 = 4 x – 12 25 = x – 12 37 = x To find YX and XZ, plug – 13 in for x and solve. 3(37) + 25 = 136 4(37) – 12 = 136

BISECTORS IN TRIANGLES Is AX the perpendicular bisector? If so, A should be equidistant from C and B. So, find the distance between the two and compare.

BISECTORS IN TRIANGLES Angle Bisector Theorem � If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. � Likewise, if a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

BISECTORS IN TRIANGLES Since D is on the angle bisector of, FD is congruent to DE.