Pre-AP Bellwork ( Bellwork sheet # 7)

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 triangles5-1 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 the distance formula. • XY 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 2x – 6 = 9 2x = 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? • 7 Find x. Find YX. Find YZ. 3x + 25 4x – 12 7

Bisectors in Triangles To find x: 3x + 25 = 4x – 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.

Pre-AP Bellwork ( Bellwork sheet # 7)