Methods 3.1 The Midpoint and Intercept theorems M3.1 The Midpoint and Intercept theorems efore you start ou should be able to: prove two triangles congruent prove two triangles similar use properties of similar triangles. Objective hy do this? rchitects have to have an excellentknowledge of geometry. Get Ready ou can understand the midpoint theorem and the intercept theorem. ou can use the midpoint theorem and the intercept theorem. 54 2x 2z a 84 x 54 2y 42 1 a The two triangles are similar explain why. b rite a in terms of x or y or z. 2 Solve these equations: a t 4 9 2 b 5 s 4 9 Key oints The Intercept theorem The lines and cut a pair of parallel lines at, and, respectively. Then: The Midpoint theorem special case of the converse of the intercept theorem is the midpoint theorem. In the triangle, and are the midpoints of the sides and. Then the line is parallel to the side of the triangle and is equal to half its length. 1
hapter 3 The Midpoint and Intercept theorems Example 1 10 cm 4. Find the length of. This problem should be solved algebraically. Using the intercept theorem: Let the length of x cm. x x 4.5 6 10 10x 6(x 4.5) x 6 4.5 4 6.75 Use the last two fractions with x and so x 4.5 Multiply through by 10(x 4.5) and cancel. Exercise 3 (ll diagrams not to scale) 1 9 cm 4 cm Find the length of. 2 12 cm 1 7 cm Find the length of. 3 Find the length of E. E 2
Methods 3.1 The Midpoint and Intercept theorems 4 2 cm 3 cm 3 cm 1 cm F E H a ork out the length of. c ork out the length of E. b ork out the length of FH. 5 is a rectangle. K, L, M and N are the midpoints of,, and respectively. Use the midpoint theorem to show that the quadrilateral KLMN is a parallelogram. 6 2 cm F E 2 cm a ork out the length of E. E b ork out the length of F. 7 kite has the property that its diagonals intersect at right angles. is a kite. E, F, G, H are the midpoints of the sides,, and respectively. rove that EFGH is a rectangle. 8 E is the midpoint of. E is parallel to. Show that: a area triangle E area triangle E b area triangle E : area triangle 1 : 2 9 is the midpoint of and is the midpoint of. and intersect at. rove that : 1 : 3 10 is a parallelogram. M is the midpoint of the diagonal and L is the midpoint of M. is the point on such that L is parallel to. is the point on such that L is parallel to. Find area L : area 3
hapter 3 The Midpoint and Intercept theorems M3.2 Intersecting hords efore you start ou should be able to: prove that two triangles are similar recall the angles in the same segment (or angles subtended at the circumference by equal arcs) theorem recall the properties of tangents to circles. Objectives hy do this? rchitects study the properties of circles so that they can design structures. Get Ready ou can solve problems which involve intersecting chords of circles. ou can solve problems which involve a chord and a tangent to a circle. 1 Show that the two triangles are similar. 2 Find the value of x. x cm 9 cm Key oints For any chords and intersecting inside a circle:. For any chords and intersecting outside a circle:. Let, and T be points on a circle. is the point of intersection of the tangent at T and the chord. T 2 T 4
Methods 3.2 Intersecting hords Example 2,, and are 4 points on a circle. and are straight lines. ork out the length of. 6 8 4.5 36 6 5.5 4. Use Example 3,, and are points on a circle. and are straight lines. 4 cm. 11 cm.. ork out the length of. 11 cm 4 cm 4 15 5 60 5 12 12 5 7 cm Use Example 4, and are 3 points on a circle. is a straight line. is a tangent to the circle at.,. Find the length of. Let x cm x(x 5) 6 2 x 2 5x 36 0 Use 2 5 x 5 Expand the brackets and collect the terms on the left hand side. (x 9)(x 4) 0 x 9 or x 4 4, 9 Select the positive value of x. Solve by factorising or using the quadratic formula. Exercise 3 (ll diagrams not to scale) 1,, and are 4 points on a circle. and are straight lines. ork out the length of. 4 cm 5
hapter 3 The Midpoint and Intercept theorems 2,, and are 4 points on a circle. is the point of intersection of the chords and. ork out the length of. 9 cm 4 cm 3,, and are points on a circle. The chords and meet inside the circle at., 12 cm,. ork out the length of. 4 opy and complete the table for the given diagram. a 9 8 6 b 10 6 4 c 8 6 20 d 4 5 29 e 5 4 15 f 4 7 19 g 12 9 25 h 9 6 24 5,, and are 4 points on a circle. 23 cm.. ork out the length of. 12 cm 6,, and are 4 points on a circle. and are straight lines. a ork out the length of. b rite down the length of. 10 cm 6
Methods 3.2 Intersecting hords 7,, and are 4 points on a circle. 10 cm. ork out the length of. 8, and are points on a circle. a Show that x satisfies the equation x 2 30x 216 b Find the value of x. 12 cm 30 cm x cm 9,, and are 4 points on a circle. The lengths of some parts of the diagram are given in the table. opy and complete the table. a 8 15 10 b 7 8 9 c 6 18 6 d 15 7 38 e 18 36 30 f 16 8 12 g 9 6 22 h 12 24 2 10 O is the centre of the circle. O is a diameter. M M. 10 cm. OM 2 cm. Find the length of M. O M 7
hapter 3 The Midpoint and Intercept theorems 11 O is the centre of the circle. O is a diameter. M is the midpoint of. The radius of the circle is c cm. OM a cm. M b cm. Show that: c 2 a 2 b 2 O M 12 O is the centre of the circle. O is a diameter. M M. M 4 cm. OM. Find the length of. O M 13 In each of the cases copy and complete the table. T a 9 16 b 4 5 T c 10 5 d 8 16 e 14 28 f 25 16 g 18 27 14, and are 3 points on a circle, centre O, radius 4.. is a straight line. is a tangent to the circle at. ork out the length of O. O 4 cm 8
Methods 3.2 Intersecting hords 15, and T are 3 points on a circle. is a straight line. T is a tangent to the circle at T. a ork out the length of T. b Given that T is a diameter of the circle, work out the length of T. c ork out the length of T. 10 cm T 16, and are 3 points on a circle. is the tangent to the circle at. is a straight line. is a diameter. rove that 2 Review T T 2 9
hapter 3 The Midpoint and Intercept theorems nswers hapter 3 M3.1 Get Ready answers 1 a Missing angles in the triangles are 84 and 42 respectively. The triangles are similar because they are equiangular. b a z 2 a t 18 b s 11.25 Exercise 3 answers 1 2 21 cm 3 1 4 a 3.2 cm b 9. c 4. 5 Join to. In triangle, KL is half of and parallel to. In triangle, MN is half of and parallel to. Hence KLMN is a parallelogram. 6 a 1.2 cm b 2.2 7 Join to. EF is parallel to, HG is parallel to. Therefore EF is parallel to HG. Similarly HE is parallel to and GF. ut is perpendicular to, so HE is perpendicular to EF and EFGH is a rectangle. 8 a E E (E is the midpoint of ) and E is common. Since the triangles have the same base and the same height they have the same area. b Triangles and have equal bases and the same height ( is common). Hence, they have the same area. Hence, rea triangle E : area triangle 1 : 2 9 is parallel to ; angle angle, angle angle. Hence triangles and are similar. Now 1 so 2 1 2 2, so 3 10 3 : 16 M3.2 Get Ready answers 1 The angles in the two triangles are equal in pairs (alternate angles and vertically opposite angles) 2 x 6.75 Exercise 3 answers 1 4 6 8 3 cm 2 9 4 36 8 4.5 12. 3 5 4 8 32 5 6.4 cm 4 a 9 8 6 12 17 18 b 10 6 4 15 16 19 c 8 12 6 16 20 22 d 4 25 5 20 29 25 e 5 12 4 15 17 19 f 21 4 12 7 25 19 g 12 12 16 9 24 25 h 12 9 6 18 21 24 5 Let x x(23 x) 5 12 23x x 2 60 x 2 23x 60 0 x 3 or x 20 20 cm 6 a 10 16 32 b 27 cm 5 7 10 6 12 2. 10 8 x(x 30 ) 12 18 x 2 30x 216 x 2 30x 216 0 (x 36)(x 6) 0, x 36 or 6. Here x 6 9 a 8 15 10 12 7 2 b 7 16 8 14 9 6 c 6 12 4 18 6 14 d 15 21 7 45 6 38 e 12 18 6 36 6 30 f 10 16 8 20 6 12 g 9 15 5 27 6 22 h 12 24 16 18 12 2 10 (5 2) (5 2) 21 M 2 21 M 21 11 (c a) (c a) M M b b So c 2 a 2 b 2 So c 2 a 2 b 2 12 M M M M (10 6) 4 M 2 M 8 2 8 2 16 2 320 cm 10
nswers 13 T a 12 9 16 7 b 6 4 9 5 c 10 5 20 15 d 8 4 16 12 e 14 7 28 21 f 15 9 25 16 g 18 12 27 15 14 2 4 (4 5) 36 6 O 2 4.5 2 6 2 56.25 O 7. 15 a 8 18 T 2 T 12 cm b T 2 18 2 12 2 T 180 cm c ( 180 ) 2 10 2 80 T 80 cm 16 2 ( ) 2 2 2 2 2 2 11