8.7 Coordinate Proof with

Transcription

1 8.7 Coordinate Proof with Quadrilaterals Goal Eample p Use coordinate geometr to prove properties of quadrilaterals. Determine if quadrilaterals are congruent Determine if the quadrilaterals with the given vertices are congruent. O(0, 0), B(2, 4), C(4, 4), D(2, 0) E(24, 0), F(22, 0), G(0, 4) H(22, 4) Graph the quadrilaterals. Show that corresponding and are congruent. Use the Distance Formula. EH 5 FG 5 OB 5 CD 5 Ï 5 EF 5 HG 5 OD 5 BC 5 Since both pairs of opposite sides in each quadrilateral are, and are parallelograms. angles in a parallelogram are congruent, so and. OB EH are, because both have slope 2, and the are cut b EO so E and O are, and E O. B substitution,. Similar reasoning can be used to show that and. Because all corresponding sides and angles congruent, OBCD congruent to EHGF. Checkpoint Find the side lengths of the quadrilaterals with the given vertices. Then determine if the quadrilaterals are congruent.. O(0, 0), B(24, 22), C(24, 6), D(22, 6) R(6, 22), S(2, 24), T(2, 4) U(4, 6) 228 Lesson 8.7 Geometr Notetaking Guide Copright Holt McDougal. All rights reserved.

2 8.7 Coordinate Proof with Quadrilaterals Goal Eample p Use coordinate geometr to prove properties of quadrilaterals. Determine if quadrilaterals are congruent Determine if the quadrilaterals with the given vertices are congruent. O(0, 0), B(2, 4), C(4, 4), D(2, 0) E(24, 0), F(22, 0), G(0, 4) H(22, 4) Graph the quadrilaterals. Show that corresponding sides and angles are congruent. Use the Distance Formula. EH 5 FG 5 OB 5 CD 5 Ï Ï 5 EF 5 HG 5 OD 5 BC 5 2 Since both pairs of opposite sides in each quadrilateral are congruent, OBCD and EHGF are parallelograms. Opposite angles in a parallelogram are congruent, so E G and O C. OB EH are parallel, because both have slope 2, and the are cut b EO so E and O are corresponding angles, and E O. B substitution, G C. Similar reasoning can be used to show that H B and F D. Because all corresponding sides and angles congruent, OBCD is congruent to EHGF.. O(0, 0), B(24, 22), C(24, 6), D(22, 6) R(6, 22), S(2, 24), T(2, 4) U(4, 6) are Checkpoint Find the side lengths of the quadrilaterals with the given vertices. Then determine if the quadrilaterals are congruent. OB 5 RS 5 Ï 20 ; BC 5 ST 5 8; CD 5 2; TU 5 Ï 8 ; DO 5 Ï 40 ; RU 5 Ï 68 ; not congruent 228 Lesson 8.7 Geometr Notetaking Guide Copright Holt McDougal. All rights reserved. E H F O G B D C

3 Eample 2 Determine if quadrilaterals are similar Determine if the quadrilaterals with the given vertices are similar. O(0, 0), B(2, 2), C(6, 2), D(4, 0) O(0, 0), F(3, 3), G(9, 3) H(6, 0) Graph the quadrilaterals. Find the ratios of corresponding side lengths. OF OB 5 3 Ï 2 2 Ï 2 CD HG 5 3 Ï 2 2 Ï FG BC OH OD Because OB 5 CD and BC 5 OD, is a. Because OF 5 HG and FG 5 OH, is a. angles in a parallelogram are, so O C and O G. Therefore C G. Parallel FG BC are cut FB, so F and CBO are and F CBO. GH DC are because both have slope, and the are cut b DH, so H and ODC are, and. Because corresponding side lengths are proportional and corresponding angles congruent, OBCD similar to OFGH. Checkpoint Determine if the quadrilaterals with the given vertices are similar. 2. O(0, 0), B(4, 22), C(4, 24), D(0, 24) O(0, 0), T(2, 22), U(, 23) V(0, 23) Copright Holt McDougal. All rights reserved. Lesson 8.7 Geometr Notetaking Guide 229

4 Eample 2 Determine if quadrilaterals are similar Determine if the quadrilaterals with the given vertices are similar. O(0, 0), B(2, 2), C(6, 2), D(4, 0) F O(0, 0), F(3, 3), G(9, 3) H(6, 0) Graph the quadrilaterals. Find the ratios of corresponding side lengths. O B D C H G OF OB 5 3 Ï 2 2 Ï 2 CD HG 5 3 Ï 2 2 Ï FG BC OH OD Because OB 5 CD and BC 5 OD, OBCD is a parallelogram. Because OF 5 HG and FG 5 OH, OFGH is a parallelogram. Opposite angles in a parallelogram are congruent, so O C and O G. Therefore C G. Parallel FG BC are cut b FB, so F and CBO are corresponding angles and F CBO. GH DC are parallel lines because both have slope, and the are cut b DH, so H and ODC are corresponding angles, and H ODC. Because corresponding side lengths are proportional and corresponding angles are congruent, OBCD is similar to OFGH. Checkpoint Determine if the quadrilaterals with the given vertices are similar. 2. O(0, 0), B(4, 22), C(4, 24), D(0, 24) O(0, 0), T(2, 22), U(, 23) V(0, 23) OBCD is not similar to OTUV. Copright Holt McDougal. All rights reserved. Lesson 8.7 Geometr Notetaking Guide 229

5 Eample 3 Demonstrate properties of quadrilaterals Show that the parallelogram is not an isosceles trapezoid. B C A D Use the Distance Formula. BD 5 and AC 5 Ï. Since the measures of the are, the trapezoid an isosceles trapezoid. Checkpoint Use properties of trapezoids. 3. What other was could ou use to show that ABCD is not an isosceles triangle? Eample 4 Determine coordinates for a verte Without introducing an new variables, suppl the missing coordinates for P so that OPQR is a parallelogram. O(0, 0) P(?,?) R(a, 0) Q(b, c) Choose coordinates so that sides of the quadrilateral are. PQ must be to be to OR, so the -coordinate of P is. To find the -coordinate of P, write epressions for the slopes of OP and RQ. Use for the of P. slope of RQ 5 c 2 5 slope of OP 5 c a The slopes are, so 5. Therefore 5. The point P has coordinates. 230 Lesson 8.7 Geometr Notetaking Guide Copright Holt McDougal. All rights reserved.

6 Eample 3 Demonstrate properties of quadrilaterals Show that the parallelogram is not an isosceles trapezoid. B C A D Use the Distance Formula. BD 5 3 Ï 5 and AC 5 Ï 34. Since the measures of the diagonals are not congruent, the trapezoid is not an isosceles trapezoid. Checkpoint Use properties of trapezoids. 3. What other was could ou use to show that ABCD is not an isosceles triangle? Show that the base angles are not congruent. Eample 4 Determine coordinates for a verte Without introducing an new variables, suppl the missing coordinates for P so that OPQR is a parallelogram. O(0, 0) P(?,?) R(a, 0) Q(b, c) Choose coordinates so that opposite sides of the quadrilateral are parallel. PQ must be horizontal to be parallel to OR, so the -coordinate of P is c. To find the -coordinate of P, write epressions for the slopes of OP and RQ. Use for the -coordinate of P. slope of RQ 5 c 2 0 c 5 b 2 a b 2 a slope of OP 5 c 2 0 The slopes are equal, so c b 2 a 5 c The point P has coordinates b 2 a, c c. Therefore b a Lesson 8.7 Geometr Notetaking Guide Copright Holt McDougal. All rights reserved.

7 Eample 5 Write a coordinate proof Prove that the diagonals of a square are perpendicular. O(0, 0) Step Place a square with side length a on the aes. Draw one side of the square from the origin to a point B(0, a) on the positive -ais. Draw the second side from B to C(a, ) within the first quadrant. Draw the third side from C to D(, ) on the positive -ais. Draw the fourth side from D back to the origin. Draw the diagonals. Step 2 Find the Slope of OC of each diagonal. Slope of BD Since the of their slopes is, the diagonals are. Checkpoint Verif that two quadrilaterals are congruent. 4. O(0, 0), S(a, 0), T(a, b), U(0, b) O(0, 0), B(2a, 0), C(2a, 2b), D(0, 2b) Homework Copright Holt McDougal. All rights reserved. Lesson 8.7 Geometr Notetaking Guide 23

8 Eample 5 Write a coordinate proof Prove that the diagonals of a square are perpendicular. B(0, a) C (a, a) O(0, 0) D(a, 0) Step Place a square with side length a on the aes. Draw one side of the square from the origin to a point B(0, a) on the positive -ais. Draw the second side from B to C(a, a) within the first quadrant. Draw the third side from C to D(a, 0) on the positive -ais. Draw the fourth side from D back to the origin. Draw the diagonals. Step 2 Find the slope of each diagonal. Slope of a 2 0 OC 5 a a a 5 Slope of BD a a a a 5 2 Since the product of their slopes is, the diagonals are perpendicular. Checkpoint Verif that two quadrilaterals are congruent. Homework 4. O(0, 0), S(a, 0), T(a, b), U(0, b) O(0, 0), B(2a, 0), C(2a, 2b), D(0, 2b) Quadrilateral OSTU and OBCD are both rectangles with horizontal and vertical sides. All angles in a rectangle are right angles, so the corresponding angles in the quadrilaterals are congruent. From the distance formula, the length of the corresponding sides are also equal: OS 5 OB 5 a, ST 5 BC 5 b, TU 5 CD 5 a, and UO 5 DO 5 b. Because corresponding angles and sides are congruent, OSTU is congruent to OBCD. Copright Holt McDougal. All rights reserved. Lesson 8.7 Geometr Notetaking Guide 23

9 Words to Review Give an eample of the vocabular word. Diagonal Parallelogram Rhombus Rectangle Square Trapezoid Bases, Legs, and Base angles of a trapezoid Isosceles trapezoid Midsegment of a trapezoid Kite Review our notes and Chapter 8 b using the Chapter Review on pages of our tetbook. 232 Words to Review Geometr Notetaking Guide Copright Holt McDougal. All rights reserved.

10 Words to Review Give an eample of the vocabular word. Diagonal Parallelogram diagonal Rhombus Rectangle Square Trapezoid Bases, Legs, and Base angles of a trapezoid Isosceles trapezoid bases legs pair of base angles Midsegment of a trapezoid Kite midsegment Review our notes and Chapter 8 b using the Chapter Review on pages of our tetbook. 232 Words to Review Geometr Notetaking Guide Copright Holt McDougal. All rights reserved.