1._ _ _ _ _ _ _ _ _ _ _ are collectively referred to as algebraic expressions.
2. If the quotient of _ _ _ _ _ _ _ _ _, then (2a+b)÷(m+n) can be expressed as _ _ _ _ _ _ _.
3. The price per kilogram of a kind of fruit is A yuan, and the price per kilogram of B kind of fruit is B yuan. Take m kilograms of class A fruit and n kilograms of class B fruit. The average price per kilogram after mixing is _ _ _ _ _ _ _.
4. Each of the following items, x+y, 3x2,0, is the score of _ _ _ _ _ _ _ _ _; Is the algebraic expression _ _ _ _ _ _ _ _? Being _ _ _ _ _ _ is rational.
5. When x _ _ _ _ _ the score is meaningless.
6. When x _ _ _ _ _ _ the value of the score is zero.
7. When x _ _ _ _ _, the value of the score is1; What is the value of the score when x _ _ _ _ _ _ 1.
8. Score, when x _ _ _ _ _ _ the score is meaningful; When x _ _ _ _ _ _ the value of the score is zero.
9. When x _ _ _ _ _ _ the value of the score is positive; When x _ _ _ _ _ the value of the score is negative.
10. If x grams of salt is dissolved in b grams of water, take out m grams of salt solution, which contains pure salt _ _ _ _ _.
1 1. Lili's distance from home to school is S. When there is no wind, she can arrive on time by bike at an average speed of one meter per second. When the wind speed is b m/s, if she arrives at school on time against the wind, please use algebra to indicate that she must leave early.
12. yongxin bottle cap factory processes a batch of bottle caps, and it takes one day for group a and group b to cooperate with each other. if group a completes it alone, it takes b days, and group b completes it alone in _ _ _ _ _ _.
13. When m = _ _ _ _ _ _ the value of the score is zero.
Multiple choice question:
14. In the following categories, no matter what value X takes, the score is meaningful ().
A.B. C. D。
15. In rational expressions ①, ②, ③ and ④, there is a fraction ().
A.①② B.③④ C.①③ D.①②③④
16. In the score, when x=-a, the following conclusion is correct ().
A. the value of the score is zero; Scores are meaningless.
C. If a ≦? , the score value is zero; D if a≦, the value of the score is zero.
Wgo5 17。 In the following categories, the possible value of zero is ().
A.B. C. D。
18. Make the score meaningless, and the value of x is ().
A.0 B. 1 C？ 1
Answer the question:
19. When x takes any value, the following scores are meaningful.
( 1) ; (2) .
20. When y= and x are known, the value of (1)y is positive; (2) the value of y is negative; (3) the value of y is zero; (4) The score is meaningless.
2 1. If the score? When the value of 1 is positive, negative and 0, find the value range of x.
Typical example
Example:
At 1. ABC, a, b and c are opposite sides of ∠A, ∠B and ∠C respectively. Under the following conditions, the number that can determine that ABC is a right triangle is ().
①a2+c2 = b2
②∠A:∠B:∠C = 1:2:3
③a:b:c = : 1: 1
④∠A:∠B:∠C = 3:4:5
A.4 B.3 C.2 D. 1
Answer: b
Note: Using the inverse theorem of Pythagorean theorem, it is easy to get ① correct; When ∠ A: ∠ B: ∠ C = 1: 2: 3, the sum of the internal angles of the triangle is 180? , it is not difficult to get ∠C is 90? , ABC is a right triangle, ② correct; When A: B: C =: 1: 1, we can set b = c = k, then a = k, then b2+c2 = 2k2 = a2, ABC is a right triangle, and ③ is correct; When ∠ A: ∠ B: ∠ C = 3: 4: 5, it is not difficult to calculate ∠A = 45? ，∠B = 60？ ，∠C = 75？ At this time, ABC is not a right triangle, and ④ is wrong; So the answer is B.
2. A triangle with the following sides A, B and C is a right triangle ().
①a =，b =，c =
②a =，b = 25，c = 24
③ A: B: C = 4:5:3.
④a = m2？ N2(m & gt； n)，b = 2mn，c = m2+n2
⑤a =，b =，c =
A.①②③④ B.①③⑤ C.②③④ D.①③④⑤
Answer: d
Description: ① It satisfies b2+c2 = a2 and is a right triangle; ②a2 = 7, b2 = 625, c2 = 576, and the B side is the longest, but it does not satisfy a2+c2 = b2, so it is not a right triangle; ③ If a = 4k, b = 5k and c = 3k, then a2+c2 = 25k2 = b2 is a right triangle; ④a2 = m4？ 2m2n2+n4, b2 = 4m2n2, c2 = m4+2m2n2+n4, satisfying a2+b2 = c2, which is a right triangle; ⑤a2 =, b2 =, c2 =, which satisfies a2+b2 = c2 and is a right triangle; So the answer is d.
3. As shown in the figure, it is known that in δδABC, ∠C = 90? , CD⊥AB at point D, let AC = b, BC = a, AB = c, CD = h；; ; Verification: ① C+H > a+b； A triangle with sides a+b, H and c+h is a right triangle.
Prove: Prove C+H > A+B. According to "Is the A>b certificate A? B& gt； There is c > in 0 ".Rt△ACB; A, c>B, so consider using an algebraic expression containing A, B and C to represent H, and get H =. That is, from S△ABC = ab = ch to prove (c+)? (a+b)>0. Use the knowledge of algebra and c > A, c> prove it.
①∫∠C = 90？ ,CD⊥AB
∴S△ABC = ab = ch
∴h = (using the triangle area formula to get the equivalent relationship)
∴(c+h)？ (a+b)
= (c+)？ (a+b)
=
= (The formula is converted into the form of product by using the grouping decomposition method in factorization)
=
∵c & gt； A>0, c>b>0 (hypotenuse of right triangle is larger than right)
∴& gt； 0
∴(c+h)？ (a+b)>0
∴c+h>； a+b
②∫c+h & gt； a+b，c+h & gt； h，a2+b2 = c2，ab = ch
∴(a+b)2+h2 = a2+B2+2ab+H2 = C2+2ch+H2 =(c+h)2
△ a+b, H and c+h are right triangles, and c+h is the hypotenuse.
4. As shown in the figure, it is known that the base BC of isosceles ABC is 20, D is a point on AB, CD = 16, BD =12; ; Find the circumference of ABC.
Solution: The length of the three sides of △BDC is 12, 16, and 20 is a set of Pythagorean numbers. It is concluded that △ ∠BDC = 90? , then △ADC is Rt△ ... Let the length of AC be X, then AD = x? 12, obtained by Pythagorean theorem (x? 12)2+ 162 = x2 to find the value of x, thus solving.
∫BD = 12，CD = 16，BC = 20，
∴bd2+cd2 = 122+ 162 = 400，BC2 = 400，
∴BD2+CD2 = BC2
∴∠BDC = 90？ , ∴△ADC is Rt△,
I'll give you some math problems, but there are no pictures. Please forgive me! Selected exercises
1. True or false
In a triangle, if the center line of one side is equal to half of this side, then the angle subtended by this side is a right angle.
(2) Proposition: "In a triangle, there is an angle of 30? Then the opposite side is half of the other side. " The counter-proposition holds.
(3) The converse theorem of Pythagorean Theorem is: If the sum of squares of two right-angled sides is equal to the square of the hypotenuse, then this triangle is a right-angled triangle.
⑷ Delta ABC is a right triangle when the ratio of three sides is 1: 1:.
Answer: right, wrong, wrong, right;
2. The opposite sides of ∠A, ∠B and ∠C in 2.△ ABC are A, B and C respectively, and the false proposition in the following proposition is ().
A. if ∠ c-∠ b = ∠ a, then △ABC is a right triangle.
B if C2 = B2-A2, then △ABC is a right triangle and ∠ c = 90.
C if (c+a) (c-a) = B2, then △ABC is a right triangle.
D if ∠ a: ∠ b: ∠ c = 5: 2: 3, then △ABC is a right triangle.
Answer: d
3. The following four line segments can't form a right triangle is ()
A.a=8，b= 15，c= 17
B.a=9，b= 12，c= 15
C.a=，b=，c=
D.a:b:c=2:3:4
Answer: d
4. It is known that in △ABC, the opposite sides of ∠A, ∠B and ∠C are A, B and C respectively, which are the following lengths. Do you judge whether a triangle is a right triangle? And point out which angle is the right angle?
⑴a=，b=，c =； ⑵a=5，b=7，c = 9；
⑶a=2，b=，c =； ⑷a=5，b=，c= 1。
Answer: (1) Yes, ∠ b; (2) no; (3) Yes, ∠ c; 4 yes, ∠ a.
5. State the inverse proposition of the latter proposition and judge whether the inverse proposition is correct.
(1) if a3 > 0, then a2 > 0；;
(2) If the angle of a triangle is less than 90, the triangle is an acute triangle;
(3) If two triangles are congruent, their corresponding angles are equal;
(4) Two line segments that are symmetrical about a straight line must be equal.
Answer: (1) If A2 > 0, A3 > 0；; False proposition.
(2) If the triangle is an acute triangle, then one angle is acute; True proposition.
(3) If the corresponding angles of two triangles are equal, then the two triangles are congruent; False proposition.
(4) Two equal line segments must be symmetrical about a straight line; False proposition.
6. Fill in the blanks.
Every proposition exists, but not every theorem exists.
The inverse theorem of "two straight lines are parallel and the internal dislocation angles are equal" is.
(3) In △ABC, if A2 = B2-C2, then △ABC is a triangle and a right angle; If A2 < B2-C2, ∠B is.
(4) If a = m2-N2, b=2mn and c = m2+N2 in △ABC, then △ABC is a triangle.
Answer: (1) inverse proposition, inverse theorem; ⑵ Internal dislocation angles are equal and two straight lines are parallel; (3) right angle, ∠B, obtuse angle; (4) right angle.
Xiao Qiang walked 80 meters to the east on the playground, then 60 meters, and then 100 meters back to his original place. Xiao Qiang walked 80 meters to the east on the playground, and then walked 60 meters.
Answer: due south or due north.
7. If three sides of a triangle are (1) 1, 2; ⑵ ; ⑶32,42,52 ⑷9,40,4 1;
⑸(m+n)2- 1，2(m+n)，(m+n)2+ 1； What constitutes a right triangle is ()
A.2 B.3 C.4 D.5
Answer: b
8. If the three sides A, B and C of △ABC satisfy (A-B) (A2+B2-C2) = 0, then △ABC is ().
A. isosceles triangle;
B. right triangle;
C. isosceles triangle or right triangle;
D. isosceles right triangle
Answer: c
9. As shown in the picture, there is a 2-meter-long shadow rod CD standing on the playground. Its shadow length BD is 4m in the morning and its shadow length AD is 1 m at noon. Can A, B and C form a right triangle? Why?
Answer: Yes, because BC2=BD2+CD2=20, AC2=AD2+CD2=5, AB2=25, BC2+AC2= AB2.
10. As shown in the picture, a ship of unknown nationality entered the offshore of China. Two patrol boats of our navy, A and B, immediately intercepted them from bases A and B, which were 13 nautical miles apart, and arrived at place C at the same time six minutes later to intercept them. It is known that patrol boat A sails at a speed of 120 nautical miles per hour, while patrol boat B sails at a speed of 50 nautical miles per hour with a heading of 40 northwest. Q
Answer: From △ABC is a right triangle, we can know that ∠ cab+∠ CBA = 90, so ∠ cab = 40, and the course is 50 northeast.
1 1. As shown in the picture, Xiaoming's father opened a quadrangular plot near the fish pond and planted some vegetables. Dad asked Xiaoming to calculate the area of the land so as to calculate the output. Xiao Ming found a roll of rice ruler and measured AB = 4m, BC = 3m, CD = 13m and DA = 12m. .
Tip: Connect AC. AC2 = AB2+BC2 = 25, AC2+AD2=CD2, so ∠CAB=90? ,
S quadrilateral =S△ADC+S△ABC=36 square meters.
12. Known in △ABC, ∠ ACB = 90, CD⊥AB in D, CD2=AD? BD。 Prove that Delta △ABC is a right triangle.
Prompt: ∫ac2 = ad2+Cd2, BC2=CD2+BD2, ∴AC2+BC2=AD2+2CD2+BD2=
AD2+2AD？ BD+BD2=(AD+BD)2=AB2,∴∠ACB=90。
13. In △ABC, AB= 13cm, AC= 24cm, BD= 5cm ... It is proved that △ABC is an isosceles triangle.
Tip: Because AD2+BD2=AB2, AD⊥BD is judged by the vertical line in the line segment, and AB = BC.
14. Known: as shown in the figure, ∠ 1=∠2, AD=AE, D is a point above BC, BD=DC, AC2 = AE2+Ce2. Verification: AB2 = AE2+Ce2.
Hint: ∠ E = 90 and AC2 = AE2+CE2；; From △ ADC △ AEC, AD=AE, CD=CE, ∠ ADC = ∠ BE = 90. Judging from the median vertical line, AB=AC, then AB2 = AE2+Ce2.
15. Given that three sides of △ABC are A, B, C, a+b=4, ab= 1, c=, try to determine the shape of △ABC.
Tip: The right triangle is proved by algebraic method, because (a+b)2= 16, a2+2ab+b2= 16, ab= 1, so A2+B2 = 14. And c2= 14.
Selected exercises
1. True or false
In a triangle, if the center line of one side is equal to half of this side, then the angle subtended by this side is a right angle.
(2) Proposition: "In a triangle, there is an angle of 30? Then the opposite side is half of the other side. " The counter-proposition holds.
(3) The converse theorem of Pythagorean Theorem is: If the sum of squares of two right-angled sides is equal to the square of the hypotenuse, then this triangle is a right-angled triangle.
⑷ Delta ABC is a right triangle when the ratio of three sides is 1: 1:.
Answer: right, wrong, wrong, right;
2. The opposite sides of ∠A, ∠B and ∠C in 2.△ ABC are A, B and C respectively, and the false proposition in the following proposition is ().
A. if ∠ c-∠ b = ∠ a, then △ABC is a right triangle.
B if C2 = B2-A2, then △ABC is a right triangle and ∠ c = 90.
C if (c+a) (c-a) = B2, then △ABC is a right triangle.
D if ∠ a: ∠ b: ∠ c = 5: 2: 3, then △ABC is a right triangle.
Answer: d
3. The following four line segments can't form a right triangle is ()
A.a=8，b= 15，c= 17
B.a=9，b= 12，c= 15
C.a=，b=，c=
D.a:b:c=2:3:4
Answer: d
4. It is known that in △ABC, the opposite sides of ∠A, ∠B and ∠C are A, B and C respectively, which are the following lengths. Do you judge whether a triangle is a right triangle? And point out which angle is the right angle?
⑴a=，b=，c =； ⑵a=5，b=7，c = 9；
⑶a=2，b=，c =； ⑷a=5，b=，c= 1。
Answer: (1) Yes, ∠ b; (2) no; (3) Yes, ∠ c; 4 yes, ∠ a.
5. State the inverse proposition of the latter proposition and judge whether the inverse proposition is correct.
(1) if a3 > 0, then a2 > 0；;
(2) If the angle of a triangle is less than 90, the triangle is an acute triangle;
(3) If two triangles are congruent, their corresponding angles are equal;
(4) Two line segments that are symmetrical about a straight line must be equal.
Answer: (1) If A2 > 0, A3 > 0；; False proposition.
(2) If the triangle is an acute triangle, then one angle is acute; True proposition.
(3) If the corresponding angles of two triangles are equal, then the two triangles are congruent; False proposition.
(4) Two equal line segments must be symmetrical about a straight line; False proposition.
6. Fill in the blanks.
Every proposition exists, but not every theorem exists.
The inverse theorem of "two straight lines are parallel and the internal dislocation angles are equal" is.
(3) In △ABC, if A2 = B2-C2, then △ABC is a triangle and a right angle; If A2 < B2-C2, ∠B is.
(4) If a = m2-N2, b=2mn and c = m2+N2 in △ABC, then △ABC is a triangle.
Answer: (1) inverse proposition, inverse theorem; ⑵ Internal dislocation angles are equal and two straight lines are parallel; (3) right angle, ∠B, obtuse angle; (4) right angle.
Xiao Qiang walked 80 meters to the east on the playground, then 60 meters, and then 100 meters back to his original place. Xiao Qiang walked 80 meters to the east on the playground, and then walked 60 meters.
Answer: due south or due north.
7. If three sides of a triangle are (1) 1, 2; ⑵ ; ⑶32,42,52 ⑷9,40,4 1;
⑸(m+n)2- 1，2(m+n)，(m+n)2+ 1； What constitutes a right triangle is ()
A.2 B.3 C.4 D.5
Answer: b
8. If the three sides A, B and C of △ABC satisfy (A-B) (A2+B2-C2) = 0, then △ABC is ().
A. isosceles triangle;
B. right triangle;
C. isosceles triangle or right triangle;
D. isosceles right triangle
Answer: c
9. As shown in the picture, there is a 2-meter-long shadow rod CD standing on the playground. Its shadow length BD is 4m in the morning and its shadow length AD is 1 m at noon. Can A, B and C form a right triangle? Why?
Answer: Yes, because BC2=BD2+CD2=20, AC2=AD2+CD2=5, AB2=25, BC2+AC2= AB2.
10. As shown in the picture, a ship of unknown nationality entered the offshore of China. Two patrol boats A and B of our navy immediately intercepted them from bases A and B, which were 13 nautical miles apart, and arrived at place C at the same time six minutes later to intercept them. It is known that patrol boat A sails at a speed of 120 nautical miles per hour, while patrol boat B sails at a speed of 50 nautical miles per hour with a heading of 40 northwest. Q
Answer: From △ABC is a right triangle, we can know that ∠ cab+∠ CBA = 90, so ∠ cab = 40, and the course is 50 northeast.
1 1. As shown in the picture, Xiaoming's father opened a quadrangular plot near the fish pond and planted some vegetables. Dad asked Xiaoming to calculate the area of the land so as to calculate the output. Xiao Ming found a roll of rice ruler and measured AB = 4m, BC = 3m, CD = 13m and DA = 12m. .
Tip: Connect AC. AC2 = AB2+BC2 = 25, AC2+AD2=CD2, so ∠CAB=90? ,
S quadrilateral =S△ADC+S△ABC=36 square meters.
12. Known in △ABC, ∠ ACB = 90, CD⊥AB in D, CD2=AD? BD。 Prove that Delta △ABC is a right triangle.
Prompt: ∫ac2 = ad2+Cd2, BC2=CD2+BD2, ∴AC2+BC2=AD2+2CD2+BD2=
AD2+2AD？ BD+BD2=(AD+BD)2=AB2,∴∠ACB=90。
13. In △ABC, AB= 13cm, AC= 24cm, BD= 5cm ... It is proved that △ABC is an isosceles triangle.
Tip: Because AD2+BD2=AB2, AD⊥BD is judged by the vertical line in the line segment, and AB = BC.
14. Known: as shown in the figure, ∠ 1=∠2, AD=AE, D is a point above BC, BD=DC, AC2 = AE2+Ce2. Verification: AB2 = AE2+Ce2.
Hint: ∠ E = 90 and AC2 = AE2+CE2；; From △ ADC △ AEC, AD=AE, CD=CE, ∠ ADC = ∠ BE = 90. Judging from the median vertical line, AB=AC, then AB2 = AE2+Ce2.
15. Given that three sides of △ABC are A, B, C, a+b=4, ab= 1, c=, try to determine the shape of △ABC.
Tip: The right triangle is proved by algebraic method, because (a+b)2= 16, a2+2ab+b2= 16, ab= 1, so A2+B2 = 14. And c2= 14.