Difference between revisions of "2019 AMC 8 Problems"
Phoenixfire (talk | contribs) (→Solution) |
Phoenixfire (talk | contribs) (→Solution) |
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== Solution == | == Solution == | ||
− | |||
− | |||
<asy> | <asy> | ||
draw((-13,0)--(0,5)); | draw((-13,0)--(0,5)); | ||
Line 64: | Line 62: | ||
draw((13,0)--(0,-5)); | draw((13,0)--(0,-5)); | ||
draw((0,-5)--(-13,0)); | draw((0,-5)--(-13,0)); | ||
− | draw((-13,0)--(13,0)); | + | draw((0,0)--(13,0)); |
− | draw((0, | + | draw((0,0)--(0,5)); |
+ | draw((0,0)--(-13,0)); | ||
+ | draw((0,0)--(0,-5)); | ||
dot((-13,0)); | dot((-13,0)); | ||
dot((0,5)); | dot((0,5)); | ||
Line 74: | Line 74: | ||
label("C",(13,0),E); | label("C",(13,0),E); | ||
label("D",(0,-5),S); | label("D",(0,-5),S); | ||
+ | label("E",(0,0),SW); | ||
</asy> | </asy> | ||
+ | |||
+ | Because it is a rhombus all sides are equal. Implies all sides are 13. In a rhombus diagonals are perpendicular and bisect each other. | ||
+ | Which means <math>\overline{AE}</math> = <math>12</math> = <math>\overline{EC}</math>. | ||
+ | |||
+ | Consider one of the right triangles. | ||
+ | |||
+ | <asy> | ||
+ | draw((-13,0)--(0,5)); | ||
+ | draw((0,0)--(-13,0)); | ||
+ | draw((0,0)--(0,5)); | ||
+ | dot((-13,0)); | ||
+ | dot((0,5)); | ||
+ | label("A",(-13,0),W); | ||
+ | label("B",(0,5),N); | ||
+ | label("E",(0,0),SE); | ||
+ | </asy> | ||
+ | |||
+ | <math>\overline{AB}</math> = <math>13</math>. | ||
+ | <math>\overline{AE}</math> = <math>12</math>. | ||
+ | Which means <math>\overline{BE}</math> = <math>5</math>. | ||
+ | |||
+ | Thus the values of the two diagonals are <math>\overline{AC}</math> = <math>24</math> and <math>\overline{BD}</math> = <math>10</math>. | ||
+ | Which means area = <math>\frac{d_1*d_2}{2}</math> = <math>\frac{24*10}{2}</math> = <math>120</math> | ||
== Problem 5 == | == Problem 5 == |
Revision as of 01:20, 20 November 2019
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Solution
- 6 Problem 5
- 7 Problem 6
- 8 Problem 7
- 9 Problem 8
- 10 Problem 9
- 11 Problem 10
- 12 Problem 11
- 13 Problem 12
- 14 Problem 13
- 15 Problem 14
- 16 Problem 15
- 17 Problem 16
- 18 Problem 17
- 19 Problem 18
- 20 Problem 19
- 21 Problem 20
- 22 Problem 21
- 23 Problem 22
- 24 Problem 23
- 25 Problem 24
- 26 Problem 25
Problem 1
Ike and Mike go into a sandwich shop with a total of to spend. Sandwiches cost each and soft drinks cost each. Ike and Mike plan to buy as many sandwiches as they can and use the remaining money to buy soft drinks. Counting both soft drinks and sandwiches, how many items will they buy?
Problem 2
Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is feet, what is the area in square feet of rectangle ? (A) 45 (B) 75 (C) 100 (D) 125 (E) 150
Problem 3
3. Which of the following is the correct order of the fractions , , and , from least to greatest?
Problem 4
Quadrilateral is a rhombus with perimeter meters. The length of diagonal is meters. What is the area in square meters of rhombus ?
Solution
Because it is a rhombus all sides are equal. Implies all sides are 13. In a rhombus diagonals are perpendicular and bisect each other. Which means = = .
Consider one of the right triangles.
= . = . Which means = .
Thus the values of the two diagonals are = and = . Which means area = = =