from: category_eng |
1. |
|
|
2. |
|
|
3. |
|
|
4. |
|
|
5. |
|
|
6. |
|
|
7. |
|
|
8. |
|
|
9. |
'
In trapezoid , and are perpendicular to , with , , and . What is ? ' |
|
10. |
'
In , we have and . Suppose that is a point on line such that lies between and and . What is ? ' |
|
11. |
'
Square has side length . A semicircle with diameter is constructed inside the square, and the tangent to the semicircle from intersects side at . What is the length of ?
|
|
12. |
'
A rectangle with a diagonal of length is twice as long as it is wide. What is the area of the rectangle? ' |
|
13. |
'
A circle of radius is surrounded by circles of radius as shown. What is ? |
|
14. |
A triangle with side lengths in the ratio is inscribed in a circle with radius 3. What is the area of the triangle?
''>''
A triangle with side lengths in the ratio is inscribed in a circle with radius 3. What is the area of the triangle? '' |
|
15. |
'
Points and lie on a circle centered at , each of and are tangent to the circle, and is equilateral. The circle intersects at . What is ? ' |
|
16. |
'
Triangle has a right angle at , , and . The bisector of meets at . What is ? ' |
|
17. |
'
A right triangle has perimeter and area . What is the length of its hypotenuse?
|
|
18. |
|
|
19. |
'
Nondegenerate has integer side lengths, is an angle bisector, , and . What is the smallest possible value of the perimeter? ' |
|
21. |
'
Point B is due east of point A. Point C is due north of point B. The distance between points A and C is , and . Point D is 20 meters due north of point C. The distance AD is between which two integers? ' |
|
1. | |
|
2. | |
|
3. | |
|
4. | |
|
5. 4 |
Complex |
|
6. 7 sqrt 2 | |
|
7. 59 |
Complex |
|
8. 15 |
Complex |
|
9. | |
By the Pythagorean theorem, we have Solving the equation, we get . |
10. | |
Draw height . We have that . From the Pythagorean Theorem, . Since , , and , so . |
11. | |
Solution 1
Solution 2Clearly, . Thus, the sides of right triangle are in arithmetic progression. Thus it is similar to the triangle and since , . |
12. | |
Let the width of the rectangle be . Then the length is Using the Pythagorean Theorem: So the area of the rectangle is |
13. | |
Solution 1 You can express the line connecting the centers of an outer circle and the inner circle in two different ways. You can add the radius of both circles to get You can also add the radius of two outer circles and use a triangle to get Since both representations are for the same thing, you can set them equal to each other. Solution 2 You can solve this problem by setting up a simple equation with the Pythagorean Theorem. The hypotenuse would be a segment that includes the radius of two circles on opposite corners and the diameter of the middle circle. This results in . The two legs are each the length between two large, adjacent circles, thus . |
14. | |
Since 3-4-5 is a Pythagorean triple, the triangle is a right triangle. Since the hypotenuse is a diameter of the circumcircle, the hypotenuse is . Then the other legs are and . The area is |
15. | |
Solution 1As is equilateral, we have , hence . Then , and from symmetry we have . Finally this gives us . We know that , as lies on the circle. From we also have , Hence , therefore , and . Solution 2As in the previous solution, we find out that . Hence and are both equilateral. We then have , hence is the incenter of , and as is equilateral, is also its centroid. Hence , and as , we have , therefore , and as before we conclude that . |
16. | |
By the Pythagorean Theorem, . The Angle Bisector Theorem now yields that |
17. | |
Solution 1Let the legs of the triangle have lengths . Then, by the Pythagorean Theorem, the length of the hypotenuse is , and the area of the triangle is . So we have the two equations Re-arranging the first equation and squaring, The length of the hypotenuse is . Solution 2From the formula , where is the area of a triangle, is its inradius, and is the semiperimeter, we can find that . It is known that in a right triangle, , where is the hypotenuse, so . Solution 3From the problem, we know that Subtracting from both sides of the first equation and squaring both sides, we get Now we substitute in as well as into the equation to get Further simplification yields the result of . Solution 4Let and be the legs of the triangle, and the hypotenuse. Since the area is 20, we have . Since the perimeter is 32, we have . The Pythagorean Theorem gives . This gives us three equations with three variables: Rewrite equation 3 as . The answer is choice (B). |
18. |
Complex |
|
19. | |
By the Angle Bisector Theorem, we know that . If we use the lowest possible integer values for AB and BC (the measures of AD and DC, respectively), then , contradicting the Triangle Inequality. If we use the next lowest values ( and ), the Triangle Inequality is satisfied. Therefore, our answer is , or choice . |
20. | |
The distance of an interior diagonal in this cube is and the distance of a diagonal on one of the square faces is . It is not possible for the fly to travel any interior diagonal twice, as then it would visit a corner more than once. So, the final sum can have at most as the coefficient of . The other 4 paths taken can be across a diagonal on one of the faces (try to find a such path!), so the maximum distance traveled is . |
21. | |
By constructing the bisectors of each angle and the perpendicular radii of the incircle the triangle consists of 3 kites. Hence and and . Let and gives three equations: (where for the first triangle.) Solving gives: Subbing in gives that has sides of . can easily be derivied from this as the sides still differ by 1 hence the above solutions still work (now with ). All additional triangles will differ by one as the solutions above differ by one so this process can be repeated indefinately until the side lengths no longer form a triangle. would have sides but these length do not make a |