Prove that when the midpoints of consecutive sides of a

Because a square is a type of rectangle, it has four right angles and its diagonals are con- gruent. Because a square is also a parallelogram, its opposite sides are parallel. For any square, we can show that the diagonals are perpendicular. In Chapter 8, we measure area in “square units.” THE RHOMBUS The next type of quadrilateral we consider is the rhombus. The plural of the word rhom- bus is rhombi pronounced rho˘m-bi¯. All sides of a rhombus are congruent. COROLLARY 4.3.4 The diagonals of a rhombus are perpendicular. THEOREM 4.3.5 A rhombus is a parallelogram with two congruent adjacent sides. DEFINITION Exs. 5–7 In Figure 4.23, the adjacent sides and of rhombus ABCD are marked con- gruent. Because a rhombus is a type of parallelogram, it is also necessary that and . Thus, we have Corollary 4.3.4. AD ⬵ BC AB ⬵ DC AD AB We will use Corollary 4.3.4 in the proof of the following theorem. D C A B Figure 4.23 EXAMPLE 2 Study the picture proof of Theorem 4.3.5. In the proof, pairs of triangles are congruent by the reason SSS. PICTURE PROOF OF THEOREM 4.3.5 D A a B E Figure 4.24 D A b B E D A c E GIVEN: Rhombus ABCD, with diagonals and [See Figure 4.24a]. PROVE: PROOF: Fold across to coincide with [see Figure 4.24b]. Now fold across half-diagonal to coincide with [see Figure 4.24c]. The four congruent triangles formed in Figure 4.24c can be unwrapped to return rhombus ABCD of Figure 4.24a. With four congruent right angles at vertex E, we see that . AC ⬜ DB 䉭AED DE 䉭CED 䉭CED AC 䉭ABC AC ⬜ DB DB AC 쮿 Geometry in the Real World The jack used in changing an automobile tire illustrates the shape of a rhombus. Discover Sketch regular hexagon RST VWX. Draw diagonals and . What type of quadrilateral is RT VX? ANSWER XV RT Rectangle An alternative definition of square is “A square is a rhombus whose adjacent sides form a right angle.” Therefore, a further property of a square is that its diagonals are perpendicular. The Pythagorean Theorem, which deals with right triangles, is also useful in appli- cations involving quadrilaterals that have right angles. In antiquity, the theorem claimed that “the square upon the hypotenuse equals the sum of the squares upon the legs of the right triangle.” See Figure 4.25a. This interpretation involves the area concept, which we study in a later chapter. By counting squares in Figure 4.25a, one sees that 25 “square units” is the sum of 9 and 16 square units. Our interpretation of the Pythagorean Theorem uses number length relationships. THE PYTHAGOREAN THEOREM The Pythagorean Theorem will be proved in Section 5.4. Although it was introduced in Section 3.2, we restate the Pythagorean Theorem here for convenience and then review its application to the right triangle in Example 3. When right angle relationships exist in quadrilaterals, we can often apply the “rule of Pythagoras” as well; see Examples 4, 5, and 6. Exs. 8–11 5 2 = 3 2 + 4 2 a 5 4 3 Figure 4.25 a c c 2 = a 2 + b 2 b b Discover How many squares are shown? 5 four 1 by 1 and one 2 by 2 ANSWER The Pythagorean Theorem In a right triangle with hypotenuse of length c and legs of lengths a and b, it follows that . c 2 = a 2 + b 2 Provided that the lengths of two of the sides of a right triangle are known, the Pythagorean Theorem can be applied to determine the length of the third side. In Example 3, we seek the length of the hypotenuse in a right triangle whose lengths of legs are known. When we are using the Pythagorean Theorem, c must represent the length of the hypotenuse; however, either leg can be chosen for length a or b. EXAMPLE 3 What is the length of the hypotenuse of a right triangle whose legs measure 6 in. and 8 in.? See Figure 4.26. Solution c 2 = 36 + 64 : c 2 = 100 : c = 10 in. c 2 = 6 2 + 8 2 c 2 = a 2 + b 2 쮿 c 8 6 Figure 4.26 c 4 3 Figure 4.27 In the following example, the diagonal of a rectangle separates it into two right tri- angles. As shown in Figure 4.27, the diagonal of the rectangle is the hypotenuse of each right triangle formed by the diagonal. EXAMPLE 4 What is the length of the diagonal in a rectangle whose sides measure 3 ft and 4 ft? Solution For each triangle in Figure 4.27, becomes or . Then , so . The length of the diagonal is 5 ft. c = 5 c 2 = 25 c 2 = 9 + 16 c 2 = 3 2 + 4 2 c 2 = a 2 + b 2 쮿 In Example 5, we use the fact that a rhombus is a parallelogram to justify that its diagonals bisect each other. By Theorem 4.3.5, the diagonals of the rhombus are also perpendicular. EXAMPLE 5 What is the length of each side of a rhombus whose diagonals measure 10 cm and 24 cm? See Figure 4.28. c 12 5 24 10 Figure 4.28 Solution The diagonals of a rhombus are perpendicular bisectors of each other. Thus, the diagonals separate the rhombus shown into four congruent right triangles with legs of lengths 5 cm and 12 cm. For each triangle, becomes , or . Then , so . The length of each side is 13 cm. c = 13 c 2 = 169 c 2 = 25 + 144 c 2 = 5 2 + 12 2 c 2 = a 2 + b 2 쮿 EXAMPLE 6 On a softball diamond actually a square, the distance along the base paths is 60 ft. Using the triangle in Figure 4.29, find the distance from home plate to second base. c 60 60 Figure 4.29 Solution Using , we have Then c = 1 7200 or c L 84.85 ft. c 2 = 7200 c 2 = 60 2 + 60 2 c 2 = a 2 + b 2 쮿 Exs. 12–14 Discover A logo is a geometric symbol that represents a company. The very sight of the symbol serves as advertising for the company or corporation. Many logos are derived from common geometric shapes. Which company is represented by these symbols? The sides of an equilateral triangle are trisected and then connected as shown, and finally the middle sections are erased. The vertices of a regular pentagon are joined to the “center” of the polygon as shown. A square is superimposed on and centered over a long and narrow parallelogram as shown. Interior line segments are then eliminated. ANSWERS Mitsubishi; Chrysler; Chevrolet When all vertices of a quadrilateral lie on a circle, the quadrilateral is a cyclic quadrilateral . As it happens, all rectangles are cyclic quadrilaterals, but no rhombus is a cyclic quadrilateral. The key factor in determining whether a quadrilateral is cyclic lies in the fact that the diagonals must intersect at a point that is equidistant from all four vertices. In Figure 4.30a, rectangle ABCD is cyclic because A, B, C, and D all lie on the circle. However, rhombus WXYZ in Figure 4.30b is not cyclic because X and Z can- not lie on the circle when W and Y do lie on the circle. A B D C a Figure 4.30 W Y X Z b EXAMPLE 7 For cyclic rectangle ABCD, . Diagonal of the rectangle is also a diameter of the circle and . Find the perimeter of ABCD shown in Figure 4.31. Solution . Let ; applying the Pythagorean Theorem with right triangle ABD, we find that . Then and , so or 6. In turn, . The perimeter of ABCD is . 28 + 26 = 16 + 12 = 28 AD = BC = 6 b = 1 36 b 2 = 36 100 = 64 + b 2 10 2 = 8 2 + b 2 AD = b AB = DC = 8 DB = 10 DB AB = 8 A B D C Figure 4.31 쮿 Exercises 4.3

1. If diagonal is congruent to each side of rhombus

ABCD , what is the measure of ? Of ? ∠ABC ∠A DB

7. A line segment joins the midpoints of two opposite sides

of a rectangle as shown. What can you conclude about and MN? MN A B D C

2. If the diagonals of a parallelogram are perpendicular, what

can you conclude about the parallelogram? HINT: Make a number of drawings in which you use only the information suggested.

3. If the diagonals of a parallelogram are congruent, what

can you conclude about the parallelogram?

4. If the diagonals of a parallelogram are perpendicular and

congruent, what can you conclude about the parallelogram?

5. If the diagonals of a quadrilateral are perpendicular

bisectors of each other but not congruent, what can you conclude about the quadrilateral?

6. If the diagonals of a rhombus are congruent, what can you

conclude about the rhombus? In Exercises 8 to 10, use the properties of rectangles to solve each problem. Rectangle ABCD is shown in the figure.

8. Given: and

Find: CD , AD, and AC not shown

9. Given: , ,

and Find: x and DA CD = 3x + 2 BC = 3x + 4 AB = 2x + 7 BC = 12 AB = 5 D M N C A B B C A D Exercises 8–10 PROOF Statements Reasons 1. Quadrilateral PQST with 1. ? midpoints A, B, C, and D of the sides 2. Draw 2. Through two points, there is one line 3. in 3. The line joining the midpoints of two sides of a triangle is to the third side 4. in 4. ? 5. 5. ? 6. Draw 6. ? 7. in 7. ? 8. in 8. ? 9. 9. ? 10. ? 10. If both pairs of opposite sides of a quadrilateral are , the quad. is a

24. Given:

Rectangle WXYZ with diagonals and Prove: ∠1 ⬵ ∠2 XZ WY ⵥ ‘ AD ‘ BC 䉭PSQ BC ‘ PS 䉭TSP AD ‘ PS PS AB ‘ DC 䉭TSQ DC ‘ TQ ‘ 䉭TPQ AB ‘ TQ TQ PROOF Statements Reasons 1. ? 1. Given 2. ? 2. The diagonals of a rectangle are 3. 3. The opposite sides of a rectangle are 4. 4. ? 5. 5. ? 6. ? 6. ?

25. Which types of quadrilaterals isare necessarily

cyclic? a A square b A parallelogram

26. Which types of quadrilaterals isare necessarily

cyclic? a A kite b A rectangle 䉭XZY ⬵ 䉭WYZ ZY ⬵ ZY ⬵ WZ ⬵ XY ⬵

10. Given: , ,

, and Find: x and y See figure for Exercise 8. In Exercises 11 to 14, consider rectangle MNPQ with diagonals and . When the answer is not a whole number, leave a square root answer.

11. If and ,

find NQ and MP.

12. If and ,

find NQ and MP.

13. If and ,

find QP and MN.

14. If and , find

MQ and NP. In Exercises 15 to 18, consider rhombus ABCD with diagonals and . When the answer is not a whole number, leave a square root answer.

15. If and ,

find AD.

16. If and ,

find AB.

17. If and ,

find AD.

18. If and , find

BC .

19. Given: Rectangle ABCD not shown with

and ; M and N are the midpoints of sides and , respectively. Find: MN

20. Given: Rhombus RSTV not shown with diagonals

and so that and Find: RS , the length of a side For Exercises 21 and 22, let , , and . Classify as true or false:

21. and

22. R

h and In Exercises 23 and 24, supply the missing statements and reasons.

23. Given: Quadrilateral PQST with midpoints A, B, C, and

D of the sides Prove: ABCD is a ⵥ R ¨ H = ⭋ H = P R 8 P H 8 P H = {rhombi} R = {rectangles} P = {parallelograms} SV = 6 RT = 8 SV RT BC AB BC = 6 AB = 8 DB = 10 AC = 14 DB = 6 AC = 10 EB = 5 AE = 6 DE = 4 AE = 5 DB AC MP = 17 QP = 15 MP = 11 NP = 7 NP = 6 QP = 9 MN = 8 MQ = 6 NQ MP DA = 3x - 3y + 1 CD = 2x - y - 1 BC = x + 2y AB = x + y Q P M N Exercises 11–14 A B E D C Exercises 15–18 T S D P Q B A C W X Z Y V 1 2 27. Find the perimeter of the cyclic quadrilateral shown. 39. A walk-up ramp moves horizontally 20 ft while rising 4 ft. Use a calculator to approximate its length to the nearest tenth of a foot.

40. a Argue that the midpoint of the hypotenuse of a right

triangle is equidistant from the three vertices of the triangle. Use the fact that the congruent diagonals of a rectangle bisect each other. Be sure to provide a drawing. b Use the relationship from part a to find CM, the length of the median to the hypotenuse of right , in which , , and .

41. Two sets of rails railroad

tracks are equally spaced intersect but not at right angles. Being as specific as possible, indicate what type of quadrilateral WXYZ is formed.

42. In square ABCD not shown, point E lies on side . If

and , find BE .

43. In square ABCD not shown, point E lies in the interior of

ABCD in such a way that is an equilateral triangle. Find . m ∠DEC 䉭ABE AE = 10 AB = 8 DC BC = 8 AC = 6 m ∠C = 90° 䉭ABC In Exercises 29 to 31, explain why each statement is true.

29. All angles of a rectangle are right angles. 30. All sides of a rhombus are congruent.

31. All sides of a square are congruent.

In Exercises 32 to 37, write a formal proof of each theorem. 32. The diagonals of a square are perpendicular. 33. A diagonal of a rhombus bisects two angles of the rhombus.

34. If the diagonals of a parallelogram are congruent, the

parallelogram is a rectangle.

35. If the diagonals of a parallelogram are perpendicular, the

parallelogram is a rhombus.

36. If the diagonals of a parallelogram are congruent and

perpendicular, the parallelogram is a square.

37. If the midpoints of the sides of a rectangle are joined in

order, the quadrilateral formed is a rhombus. In Exercises 38 and 39, you will need to use the square root function of your calculator.

38. A wall that is 12 ft long by 8 ft high has a triangular brace

along the diagonal. Use a calculator to approximate the length of the brace to the nearest tenth of a foot. 1

28. Find the perimeter of the square shown.

A B C D 39 52 25 3 冑 2 8 12 20 4 W X Y Z