Bisectors in Triangles Lesson 5-2 Lesson 5-1 Quiz – Midsegments of Triangles In GHI, R, S, and T are midpoints. 1. Name all the pairs of parallel segments. 2. If GH = 20 and HI = 18, find RT. 3. If RH = 7 and RS = 5, find ST. 4. If m G = 60 and m I = 70, find m GTR. 5. If m H = 50 and m I = 66, find m ITS. 6. If m G = m H = m I and RT = 15, find the perimeter of GHI. RT || HI, RS || GI, ST || HG 9 7 70 64 90 5-1

When a point is the same distance from two or more Bisectors in Triangles Lesson 5-2 When a point is the same distance from two or more objects, the point is said to be equidistant from the objects. Triangle congruence theorems can be used to prove theorems about equidistant points. 5-2

Bisectors in Triangles Lesson 5-2 5-2

Bisectors in Triangles Lesson 5-2 5-2

Bisectors in Triangles Lesson 5-2 The shortest segment from a point to a line is perpendicular to the line. This fact is used to define the distance from a point to a line as the length of the perpendicular segment from the point to the line. 5-2

Bisectors in Triangles Lesson 5-2 5-2

Bisectors in Triangles Lesson 5-2 Additional Examples Real-World Connection Use the map of Washington, D.C. Describe the set of points that are equidistant from the Lincoln Memorial and the Capitol. The Converse of the Perpendicular Bisector Theorem states If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Quick Check 5-2

Bisectors in Triangles Lesson 5-2 Additional Examples (continued) A point that is equidistant from the Lincoln Memorial and the Capitol must be on the perpendicular bisector of the segment whose endpoints are the Lincoln Memorial and the Capitol. Therefore, all points on the perpendicular bisector of the segment whose endpoints are the Lincoln Memorial and the Capitol are equidistant from the Lincoln Memorial and the Capitol. Quick Check 5-2

Bisectors in Triangles Lesson 5-2 Additional Examples Using the Angle Bisector Theorem Find x, FB, and FD in the diagram above. FD = FB Angle Bisector Theorem 7x – 37 = 2x + 5 Substitute. 7x = 2x + 42 Add 37 to each side. 5x = 42 Subtract 2x from each side. x = 8.4 Divide each side by 5. FB = 2(8.4) + 5 = 21.8 Substitute. FD = 7(8.4) – 37 = 21.8 Substitute. Quick Check 5-2

Bisectors in Triangles Lesson 5-2 Lesson Quiz Use this figure for Exercises 1–3. 1. Find BD. 2. Complete the statement: C is equidistant from ? . 3. Can you conclude that CN = DN? Explain. Use this figure for Exercises 4–6. 4. Find the value of x. 5. Find CG. 6. Find the perimeter of quadrilateral ABCG. 16 6 8 points A and B No; if CN = DN, CNB DNB by SAS and CB = DB by CPCTC, which is false. 48 5-2

Bisectors in Triangles Lesson 5-2 Check Skills You’ll Need 1. Draw a triangle, XYZ. Construct STV so that STV XYZ. 2. Draw acute  P. Construct  Q so that 3. Draw AB. Construct a line CD so that CD AB and CD bisects AB. 4. Draw acute angle  E. Construct the bisector of  E. TM bisects  STU so that m STM = 5x + 4 and m MTU = 6x – 2. 5. Find the value of x. 6. Find m STU. (For help, go to Lesson 1-7.) Use a compass and a straightedge for the following.  Q  P. Check Skills You’ll Need 5-2

Bisectors in Triangles Lesson 5-2 Check Skills You’ll Need Solutions 1-4. Answers may vary. Samples given: 1. 2. 3. 4. 5. Since TM bisects  STU, m STM = m MTU. So, 5x + 4 = 6x – 2. Subtract 5x from both sides: 4 = x – 2; add 2 to both sides: x = 6. 6. From Exercise 5, x = 6. m STU = m STM + m MTU = 5x + 4 + 6x – 2 = 11x + 2 = 11(6) + 2 = 68. 5-2