Geometry
Katie glues a rectangular picture measuring 9 cm by 26 cm to a rectangular piece of paper so that there is 8 cm of paper showing on each side of the picture. What is the total area, in square centimeters, of the paper that is showing around the edges of the picture?
The perimeter of the picture is 9 + 26 + 9 + 26 = 35 + 35 = 70, and there is a rectangle extending 8 cm from each side, so the combined areas of these rectangles will be 8 ⋅ 70 = 560. In addition there are four 8 cm squares in the corners, with a combined area of 4 x 82 = 162 = 256, giving an answer of 560 + 256 = 816.
The solution, in the form (k, m), of the system of equations
k+4m=-31
k+2m=-17
What is (-3,-7)?
Subtracting the second equation from the first gives 2m=-14, so m=-14/2=-7, substituting gives you m=-3
How many complete months are in 36 years?
There are 12 months per year, for an answer of 12 ⋅ 36 = 144 ⋅ 3 = 432.
Express in simplest radical form: 5√96
5√96 = 5√(25 x 3)= 2 5√3
A right triangle has a hypotenuse measuring 39 ft and one leg measuring 36 ft. What is the length, in feet, of the other leg?
This triangle is 3 times as big a triangle with a hypotenuse of 13 and a leg of 12, which you may recognize gives another leg of 5, for an answer of 3 ⋅ 5 = 15. If you did not recognize the Pythagorean triple, you could use the Pythagorean Theorem.
What is the name for a polygon with eight sides?
You should have memorized “octagon”.
The equation, in slope-intercept (y=mx+b) of the line through the points (-5, -7) and (9, 35)
What is y=3x+8?
How many dollars are 2 quarters, 3 dimes, 9 nickels, and 4 pennies worth? For example, your answer might be 7.38.
2 x 0.25 + 3 x 0.1 + 9 x 0.05 + 4 x 0.01 = 0.50 + 0.30 + 0.45 + 0.04 = 1.29
Evaluate: 652 − 552
652 − 552 = (65 − 55)(65 + 55) = 10 ⋅ 120 = 1200
What is the area, in square meters, of a right triangle with legs measuring 5 m and 14 m?
A right triangle is half a rectangle (diagonally), so its area is
Two similar rhombi have perimeters of 60 m and 40 m. If the smaller rhombus has an area of 30 m2 , what is the area, in square meters, of the larger rhombus?
The areas will be in the ratio of ( 60/40)2 = ( 6/4)2 = (3/2)2 =9/4 , for an answer of 9/4 x 30= 9/2 x 15 = 135/2 .
The coordinates, in the form (x,y), of the line segment connecting (-9, 6) and (2, -3)
What is (− 7/ 2 , 3/ 2 )?
The midpoint will be at ( −9+2/ 2 , 6+(−3)/ 2 ) = (− 7/ 2 , 3/ 2 ).
If today is Thursday, what day of the week was it 567 days ago?
7 days ago was Thursday, as was 70, as would be 700; it’s all about 7s. 567 = 7 ⋅ 81, so 567 days ago was exactly 8 weeks ago, so it was Thursday.
Simplify: 4/√2
4/√2 = 4√2/2 = 2√2
What is the length, in feet, of a leg of a right triangle with one angle measuring 45 degrees and a hypotenuse measuring 8 feet?
In a 45-45-90 triangle, the hypotenuse is √2 times each leg, for an answer of 8/√2 = 8√2 /2 = 4√2.
How many vertices does a regular dodecahedron have?
There are five around the “top” pentagon, five around the “bottom” pentagon, and ten around the “middle”, for a total of 5 + 5 + 10 = 20.
Equation of line drawn on the board.
A. 14x-8y=6 B. 9x+8y=7
C. 8x-7y=-6 D. 2x+4y=-3
What is C?
The line has a positive y-intercept, which could be either B or C. The line also has a negative x-intercept, which limits the options to C.
When the secret number is reduced by 35 and this result is divided by 9, the final result is 29. What is the secret number?
If the final result was 29, the intermediate result must have been 29 ⋅ 9 = 261, so that the secret number is 261 + 35 = 296.
Evaluate: log232
Because 25 = 32, log232 = 5.
What is the most specific name for a triangle with two angles measuring 54° and 49°
The third angle is 180 − 54 − 49 = 180 − 103 = 77°. The three angles are different, so the three sides will be different, so the triangle is “scalene”. In addition, the three angles are all less than 90°, so the triangle is “acute”
What is the perimeter, in meters, of a square that is inscribed in a circle with a circumference of 7
28/pi
The coordinates, in the form (x,y), of the vertex of the parabola with equation y=7x^2+126x-49
What is (−9, −616)?
The vertex will lie on the axis of symmetry, which is x=-b/2a=-126/(2⋅7)=-63/7=-9, so its y-value will be y=7(-9)^2+126(-9)-49=-7(81)-49=-567-49=-616, for an answer of (-9,-616).
In a game of Hide & Seek, Pete and Tink see one another when they are 72 feet apart, and the chase is on! If Pete chases Tink at a speed of 6 feet per second, and Tink runs away at a speed of 2 feet per second, how many seconds will it take Pete to catch Tink?
Pete can close the distance at a rate of 6 − 2 = 4 fps, so it will take him 72 4 = 18 seconds to catch Tink.
Express the base 10 number 699 as a base 7 number
In base seven, the digits from right to left represent 70= 1, 71 = 7, 72 = 49, 73= 343, etc. 699 contains two 343s, leaving 699 − 686 = 13, so there are no 49s, one 7, and six 1s, for an answer of 2016.
What is the smallest possible perimeter, in meters, of an isosceles triangle with sides measuring 6 m and 17 m?
You can’t have a 6-6-17 triangle, because 6 + 6 = 12 < 17, so it must be a 6-17-17 triangle with a perimeter of 6 + 17 + 17 = 40.