Download Geometry Hustle Solutions

Geometry Hustle Solutions
1. Because the diagonals of a rhombus intersect at right angles, the side and half the diagonals
must form a Pythagorean triple (12,35,37). The area of a rhombus is half the product of the
diagonals: 24*35 = 840
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2. We need to use the formula Angle = 2 |60ℎ − 11𝑚| where the angle is the obtuse angle
between the two hands. So angle = 360-137.5=225.5 ,and we use h=12. Plugging this into the
formula gives you m=25.
3. You can see that the side length of the square is 6. So, its area is 36. The square also has a
diagonal length of 6√2 which means the radius of the outer circle is 3√2. Area of a circle is 𝐴 =
𝜋𝑟 2 = (3√2)2 𝜋 = 18π. Area of outer circle minus area of square 18π-36.
4. The conic is an ellipse with miner or major axis of 5 and 7. The area is thus 35π
5. Assume a distance of d between the two poles. The height of the intersection is x. Drawing a line
perpendicular to the ground, going through the intersection point (note, this is shown in the
drawing, with length x), we see that d is cut into two sections. Because the poles and
intersecting wire forms similar triangles, these sections must be in the ratio 5:3. Consider the
section closer to the 5 ft pole – its length must be 5d/8. Next, consider the line connecting the
bottom of the 5ft pole to the top of the 3ft pole. The intersection again separates this wire into
a ratio of 5:3. Call this length w, and the wire connecting the bottom of the 5ft pole to the
5𝑑
8
5𝑤 2
)
8
intersection is thus 5w/8 in length. We now note that 𝑤 2 + 32 = 𝑑2 and ( )2 + 𝑥 2 = (
6.
7.
8.
9.
Solve these equations to get 15/8
Flatland (Flatland: A Romance of Many Dimensions)
Each angle takes a value of (360*3/(2*7)), which can be seen if one draws the picture. Thus the
sum is 540 degrees.
Each Peeli must shake hands twice, so we can think of this as coupling them, twice: 12C2*2 = 132
handshakes.
B. A radius and a point define a sphere.
5
𝑥
10. Use similar triangles to set up the ratio 12 = 𝑥+8 where x is the length of his shadow. Solving for
x gives you 40/7 feet.
11. Use law of cosines to find c, where 𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝜃, and a and b are the sides of the
triangle. plug in to get c = 2√13.
12. Incenter
13. Using Decartes’ theorem, we know that the curvature of the 4th circle must be equal to 1+4+4 +
or – 6. This gives a possible radius of 1/12 or infinity. The radius of infinity corresponds to a line,
which is not a circle. Thus there is only one possible radius 1.
14. With the exterior angle equal to 400, the interior will be 180-40 or 140o. To find the number of
sides we do 140 =
180(𝑛−2)
.
𝑛
Solving this gives you n=9.
15. One can form a right triangle using half the tangent and the radius of the inner circle – 5 and 12.
The hypotenuse of this triangle (also the radius of the outer circle) is thus 13. So, (132 − 122 ) ∗
𝜋 = 25𝜋
Geometry Hustle Solutions
16. There exist 5 platonic solids: tetrahedron (4 faces), hexahedron (6 faces), octahedron (8 faces),
dodecahedron (12 faces), and icosahedron (20 faces). So the total number is 4+6+8+12+20 = 50
faces.
17. This is a cyclic quadrilateral whose area we can find with Brahmagupta's formula 𝐴 =
√(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)(𝑠 − 𝑑), where s = semiperimeter, and a,b,c,d are the side lengths of
the quadrilateral. So, here s = 13, and thus, area = 4√105.
18. The pink area = 150. The total area of the brown = ([5*5*5]*[1*1]*6) = 750. 150/750 *100 = 20
or 20%
19. To find the longest diagonal, use the formula √𝑙 2 + 𝑤 2 + ℎ2 . So, diagonal = √52 + 72 + 92 =
√155.
20. If one angle is equal to 90, all other angles (supplementary or similar angles) must also be 90
degrees.
21. If the base areas of the two cones have a ratio of 441:pi, then the ratio of their bases is 21:√𝜋.
3
Thus the ratio of their volumes must be 9261:𝜋 ⁄2
22. From two right triangles and a rectangle and break up the bottom side into lengths of 4, 12, and
4. You'll thus see that the height is 3 and so find the area of the 2 triangles and rectangle to get
48.
23. The third side must be shorter than (13 +7) = 20, and longer than (13 – 7) = 6. There are only 13
integers that satisfy this condition.
10!
24. This statement is untruthful: 3!2! = 302400
25. The centroid is simply found by averaging the coordinates of the three points: (6,3)