Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: The shaded part of the regular octagon has an area of $3 \mathrm{~cm}^{2}$. How big is the area of the octagon?
<image1>
Choices:
A. $8+4 \sqrt{2} \mathrm{~cm}^{2}$
B. $9 \mathrm{~cm}^{2}$
C. $8 \sqrt{2} \mathrm{~cm}^{2}$
D. $12 \mathrm{~cm}^{2}$
E. $14 \mathrm{~cm}^{2}$
Answer: D
Problem: <image>Question: If you add the numbers on opposite faces of this special die, you will get the same total three times. The numbers on the hidden faces of the die are prime numbers. Which number is on the face opposite to 14?
<image1>
Answer: 23
Problem: <image>Question: In the diagram Karl wants to add lines joining two of the marked points at a time, so that each of the seven marked points is joined to the same number of other marked points. What is the minimum number of lines he must draw?
<image1>
Answer: 9
Problem: <image>Question: The diagram shows two different views of the same cube. The cube is made from 27 small cubes, which are either white or black. At most how many black cubes are there?
<image1>
Answer: 9
Problem: <image>Question: $P T$ is the tangent to a circle $O$, and $P B$ is the angle bisector of the angle TPA (see diagram). How big is the angle TBP?
<image1>
Choices:
A. $30^{\circ}$
B. $45^{\circ}$
C. $50^{\circ}$
D. $75^{\circ}$
E. It depends on the location of point $P$
Answer: B
Problem: <image>Question: In triangle $A B C, A B=6 \mathrm{~cm}, A C=8 \mathrm{~cm}$ and $B C=10 \mathrm{~cm}$. $M$ is the midpoint of the side $B C$. $A M D E$ is a square and $M D$ intersects $A C$ at point $F$. What is the area of the quadrilateral $A F D E$ in $\mathrm{cm}^{2}$?
<image1>
Choices:
A. $\frac{124}{8}$
B. $\frac{125}{8}$
C. $\frac{126}{8}$
D. $\frac{127}{8}$
E. $\frac{128}{8}$
Answer: B
Problem: <image>Question: The grey areas of the square with side length $a$ are bounded by a semi-circle and two quarter-circles respectively. What is their total area?
<image1>
Choices:
A. $\frac{\pi a^{2}}{8}$
B. $\frac{a^{2}}{2}$
C. $\frac{\pi a^{2}}{2}$
D. $\frac{a^{2}}{4}$
E. $\frac{\pi a^{2}}{4}$
Answer: B
Problem: <image>Question: Mr. Hide wants to find a treasure that he buried years before. He can only remember that he buried the treasure at least $5 \mathrm{~m}$ away from the hedge and no more than $5 \mathrm{~m}$ away from the old pear tree. Which picture shows best the area where Mr. Hide has to look for the treasure?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: In the diagram one can see my decision-die in three different positions. What is the probability I get a "YES", when rolling this die once.
<image1>
Choices:
A. $\frac{1}{3}$
B. $\frac{1}{2}$
C. $\frac{5}{9}$
D. $\frac{2}{3}$
E. $\frac{5}{6}$
Answer: B
Problem: <image>Question: The side lengths of each of the small squares in the diagram are 1. How long is the shortest path from "Start" to "Ziel", if you are only allowed to move along the sides and the diagonals of the squares?
<image1>
Choices:
A. $2 \sqrt{5}$
B. $\sqrt{10}+\sqrt{2}$
C. $2+2 \sqrt{2}$
D. $4 \sqrt{2}$
E. 6
Answer: C
Problem: <image>Question: The square $A B C D$ has area 80. The points $E, F, G$ and $H$ are on the sides of the square and $\mathrm{AE}=\mathrm{BF}=\mathrm{CG}=\mathrm{DH}$. How big is the area of the grey part, if $\mathrm{AE}=3 \times \mathrm{EB}$?
<image1>
Answer: 25
Problem: <image>Question: Four objects $a, b, c, d$ are placed on a double balance as shown. Then two of the objects are exchanged, which results in the change of position of the balance as shown. Which two objects were exchanged?
<image1>
Choices:
A. $a$ and $b$
B. $b$ and $d$
C. $b$ and $c$
D. $a$ and $d$
E. $a$ and $c$
Answer: D
Problem: <image>Question: In the diagram we can see seven sections which are bordered by three circles. One number is written into each section. It is known that each number is equal to the sum of all the numbers in the adjacent zones. (Two zones are called adjacent if they have more than one corner point in common.) Which number is written into the inner area?
<image1>
Answer: 0
Problem: <image>Question: Lines parallel to the base $A C$ of triangle $A B C$ are drawn through $X$ and $Y$. In each case, the areas of the grey parts are equal in siz. The ratio $B X: X A=4: 1$ is known. What is the ratio $B Y: Y A$?
<image1>
Choices:
A. $1: 1$
B. $2: 1$
C. $3: 1$
D. $3: 2$
E. $4: 3$
Answer: D
Problem: <image>Question: A $3 \times 3$ field is made up of 9 unit squares. In two of these squares, circles are inscribed as shown in the diagram. How big is the shortest distance between these circles?
<image1>
Choices:
A. $2 \sqrt{2}-1$
B. $\sqrt{2+1}$
C. $2 \sqrt{2}$
D. 2
E. 3
Answer: A
Problem: <image>Question: What percentage of the area of the triangle is coloured in grey in the adjacent diagram?
<image1>
Choices:
A. $80 \%$
B. $85 \%$
C. $88 \%$
D. $90 \%$
E. It cannot be calculated.
Answer: C
Problem: <image>Question: Jilly makes up a multiplication magic square using the numbers $1,2,4,5,10,20,25,50$ and 100. The products of the numbers in each row, column and diagonal should be equal. In the diagram it can be seen how she has started. Which number goes into the cell with the question mark?
<image1>
Answer: 4
Problem: <image>Question: Jack wants to keep six tubes each of diameter $2 \mathrm{~cm}$ together using a rubber band. He chooses between the two possible variations shown. How are the lengths of the rubber bands related to each other?
<image1>
Choices:
A. In the left picture the band is $\pi \mathrm{cm}$ shorter.
B. In the left picture the band is $4 \mathrm{~cm}$ shorter.
C. In the right picture the band is $\pi \mathrm{cm}$ shorter.
D. In the right picture the band is $4 \mathrm{~cm}$ shorter.
E. Both bands are equally long.
Answer: E
Problem: <image>Question: Peter wants to colour in the cells of a $3 \times 3$ square so that every row, every column and both diagonals each have three cells with three different colours. What is the smallest number of colours with which Peter can achieve this?
<image1>
Answer: 5
Problem: <image>Question: In the diagram we see a cube and four marked angles. How big is the sum of those angles?
<image1>
Choices:
A. $315^{\circ}$
B. $330^{\circ}$
C. $345^{\circ}$
D. $360^{\circ}$
E. $375^{\circ}$
Answer: B
Problem: <image>Question: A creeping plant twists exactly 5 times around a post with circumference $15 \mathrm{~cm}$ (as shown in the diagram) and thus reaches a height of $1 \mathrm{~m}$. While the plant grows the height of the plant also grows with constant speed. How long is the creeping plant?
<image1>
Choices:
A. $0.75 \mathrm{~m}$
B. $1.0 \mathrm{~m}$
C. $1.25 \mathrm{~m}$
D. $1.5 \mathrm{~m}$
E. $1.75 \mathrm{~m}$
Answer: C
Problem: <image>Question: We consider a $5 \times 5$ square that is split up into 25 fields. Initially all fields are white. In each move it is allowed to change the colour of two fields that are horizontally or vertically adjacent (i.e. white fields turn black and black ones turn white). What is the smallest number of moves needed to obtain the chessboard colouring shown in the diagram?
<image1>
Answer: 12
Problem: <image>Question: Peter writes the word KANGAROO on a see-through piece of glass, as seen on the right. What can he see when he first flips over the glass onto its back along the right-hand side edge and then turns it about $180^{\circ}$ while it is lying on the table?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: A wheel rolls along a zigzag curve as can be seen below. Which of the following pictures shows the curve that is described by the centre of the wheel?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: A circle with radius 1 rolls along a straight line from point $K$ to point $L$, as shown, with $K L=11 \pi$. In which position is the circle when it has arrived in $L$?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: $A B C D$ is a trapezium with parallel sides $A B$ and $C D$. Let $A B=50$ and $C D=20$. Point $E$ lies on side $A B$ in such a way that the straight line $D E$ divides the trapezium into two shapes of equal area. How long is the straight line $A E$?
<image1>
Answer: 35
Problem: <image>Question: In an equilateral triangle with area 1, we draw the six perpendicular lines from the midpoints of each side to the other two sides as seen in the diagram. How big is the area of the grey hexagon that has been created this way?
<image1>
Choices:
A. $\frac{1}{3}$
B. $\frac{2}{5}$
C. $\frac{4}{9}$
D. $\frac{1}{2}$
E. $\frac{2}{3}$
Answer: D
Problem: <image>Question: A belt system is made up of wheels $A, B$ and $C$, which rotate without sliding. $B$ rotates 4 times around, while $A$ turns 5 times around, and $B$ rotates 6 times around, while $C$ turns 7 times around. The circumference of $C$ is $30 \mathrm{~cm}$. How big is the circumference of $A$?
<image1>
Choices:
A. $27 \mathrm{~cm}$
B. $28 \mathrm{~cm}$
C. $29 \mathrm{~cm}$
D. $30 \mathrm{~cm}$
E. $31 \mathrm{~cm}$
Answer: B
Problem: <image>Question: Jenny wants to write numbers into the cells of a $3 \times 3$-table so that the sum of the numbers in each of the four $2 \times 2$-squares are equally big. As it is shown in the diagram, she has already inserted three numbers. What number does she have to write into the cell in the fourth corner?
<image1>
Answer: 0
Problem: <image>Question: In a convex quadrilateral $A B C D$ the diagonals are perpendicular to each other. The length of the edges are $A B=2017, B C=2018$ and $C D=2019$ (diagram not to scale). How long is side $A D$?
<image1>
Choices:
A. 2016
B. 2018
C. $\sqrt{2020^{2}-4}$
D. $\sqrt{2018^{2}+2}$
E. 2020
Answer: D
Problem: <image>Question: Paul wants to write a positive whole number onto every tile in the number wall shown, so that every number is equal to the sum of the two numbers on the tiles that are directly below. What is the maximum number of odd numbers he can write on the tiles?
<image1>
Answer: 14
Problem: <image>Question: Three weights are randomly placed on each tray of a beam balance. The balance dips to the right hand side as shown on the picture. The masses of the weights are 101, 102, 103, 104, 105 and 106 grams. For how many percent of the possible distributions is the 106grams-weight on the right (heavier) side?
<image1>
Choices:
A. $75 \%$
B. $80 \%$
C. $90 \%$
D. $95 \%$
E. $100 \%$
Answer: B
Problem: <image>Question: The points $A$ and $B$ lie on a circle with centre $M$. The point $P$ lies on the straight line through $A$ and $M. P B$ touches the circle in $B$. The lengths of the segments $P A$ and $M B$ are whole numbers, and $P B=P A+6$. How many possible values for $M B$ are there?
<image1>
Answer: 6
Problem: <image>Question: The rings shown are partially interlinked. How long is the longest chain built this way which also contains the thick light ring?
<image1>
Answer: 5
Problem: <image>Question: The distance between the top of the cat that is sitting on the table to the top of the cat that is sleeping on the floor is $150 \mathrm{~cm}$. The distance from the top of the cat that is sleeping on the table to the top of the cat that is sitting on the floor is $110 \mathrm{~cm}$. How high is the table?
<image1>
Choices:
A. $110 \mathrm{~cm}$
B. $120 \mathrm{~cm}$
C. $130 \mathrm{~cm}$
D. $140 \mathrm{~cm}$
E. $150 \mathrm{~cm}$
Answer: C
Problem: <image>Question: In the three regular hexagons shown, $X, Y$ and $Z$ describe in this order the areas of the grey shaded parts. Which of the following statements is true?
<image1>
Choices:
A. $X=Y=Z$
B. $Y=Z \neq X$
C. $Z=X \neq Y$
D. $X=Y \neq Z$
E. Each of the areas has a different value.
Answer: A
Problem: <image>Question: In the (correct) calculation shown, some of the digits were replaced by the letters P, Q, R and S. What is the value of $P+Q+R+S$?
<image1>
Answer: 15
Problem: <image>Question: In the diagram shown, you should follow the arrows to get from A to B. How many different ways are there that fulfill this condition?
<image1>
Answer: 16
Problem: <image>Question: Eight congruent semi-circles are drawn inside a square with side length 4. How big is the area of the white part?
<image1>
Answer: 8
Problem: <image>Question: Two concentric circles with radii 1 and 9 form an annulus. $n$ circles without overlap are drawn inside this annulus, where every circle touches both circles of the annulus. (The diagram shows an example for $\mathrm{n}=1$ and the other radii as given.) What is the biggest possible value of $n$?
<image1>
Answer: 3
Problem: <image>Question: A number is to be written into every vertex of the 18 -sided shape so that it is equal to the sum of the two numbers from the adjacent vertices. Two of these numbers are given. Which number is written in vertex $A$?
<image1>
Answer: 38
Problem: <image>Question: Diana draws a rectangle made up of twelve squares onto a piece of squared paper. Some of the squares are coloured in black. She writes the number of adjacent black squares into every white square. The diagram shows an example of such a rectangle. Now she does the same with a rectangle made up of 2018 squares. What is the biggest number that she can obtain as the sum of all numbers in the white squares?
<image1>
Answer: 3025
Problem: <image>Question: Seven little dice were removed from a $3 \times 3 \times 3$ die, as can be seen in the diagram. The remaining (completely symmetrical) figure is cut along a plane through the centre and perpendicular to one of the four space diagonals. What does the cross-section look like?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: Two chords $A B$ and $A C$ are drawn into a circle with diameter $\mathrm{AD}. \angle B A C=60^{\circ}$, $\overline{A B}=24 \mathrm{~cm}$, $\mathrm{E}$ lies on $\mathrm{AC}$ so that $\overline{E C}=3 \mathrm{~cm}$, and $\mathrm{BE}$ is perpendicular to $\mathrm{AC}$. How long is the chord $\mathrm{BD}$?
<image1>
Choices:
A. $\sqrt{3} \mathrm{~cm}$
B. $2 \mathrm{~cm}$
C. $3 \mathrm{~cm}$
D. $2 \sqrt{3} \mathrm{~cm}$
E. $3 \sqrt{2} \mathrm{~cm}$
Answer: D
Problem: <image>Question: A barber wants to write the word SHAVE on a board so that a customer who sees the word in the mirror can read the word normally. How does he have to write the word on the board?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: Five identical measuring jugs are filled with water. Four of them contain exactly the same amount of water. Which measuring jug contains a different amount?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: Which of the following statements is definitely true for the angle marked in the diagram which is made up of nine squares?
<image1>
Choices:
A. $\alpha=\beta$
B. $2 \alpha+\beta=90^{\circ}$
C. $\alpha+\beta=60^{\circ}$
D. $2 \beta+\alpha=90^{\circ}$
E. $\alpha+\beta=45^{\circ}$
Answer: B
Problem: <image>Question: Inside a unit square a certain area has been coloured in black. In which square is the black area biggest?
<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: A
Problem: <image>Question: Three five-digit numbers are written onto three separate pieces of paper as shown. Three of the digits in the picture are hidden. The sum of the three numbers is 57263. Which are the hidden digits?
<image1>
Choices:
A. 0,2 and 2
B. 1,2 and 9
C. 2,4 and 9
D. 2,7 and 8
E. 5,7 and 8
Answer: B
Problem: <image>Question: The flag of Kanguria is a rectangle whose side lengths are in the ratio $3: 5$. The flag is split into four rectangles of equal area as shown. In which ratio are the side lengths of the white rectangle?
<image1>
Choices:
A. $1: 3$
B. $1: 4$
C. $2: 7$
D. $3: 10$
E. $4: 15$
Answer: E
Problem: <image>Question: A $3 \times 2$ rectangle can be covered in two ways by two of the L-shaped figures as shown:
<image1>
In how many ways can the diagram below be covered by these L-shaped figures?
<image2>
Answer: 2
Problem: <image>Question: The diagram consists of three circles of equal radius $R$. The centre of those circles lie on a common straight line where the middle circle goes through the centres of the other two circles (see diagram). How big is the perimeter of the figure?
<image1>
Choices:
A. $\frac{10 \pi R}{3}$
B. $\frac{5 \pi R}{3}$
C. $\frac{2 \pi R \sqrt{3}}{3}$
D. $2 \pi R \sqrt{3}$
E. $4 \pi R$
Answer: A
Problem: <image>Question: The diagram shows the net of an octahedron. Which edge meets the edge labelled with $\mathrm{x}$ if the net is folded up to form an octahedron?
<image1>
Answer: 5
Problem: <image>Question: Two vertices of a square lie on a semi-circle as shown, while the other two lie on its diameter. The radius of the circle is $1 \mathrm{~cm}$. How big is the area of the square?
<image1>
Choices:
A. $\frac{4}{5} \mathrm{~cm}^{2}$
B. $\frac{\pi}{4} \mathrm{~cm}^{2}$
C. $1 \mathrm{~cm}^{2}$
D. $\frac{4}{3} \mathrm{~cm}^{2}$
E. $\frac{2}{\sqrt{3}} \mathrm{~cm}^{2}$
Answer: A
Problem: <image>Question: A graph consists of 16 points and several connecting lines as shown in the diagram. An ant is at point $A$. With every move the ant can move from the point where it currently is, along one of the connecting lines, to an adjacent point. At which of the points $P, Q, R, S$ and $T$ can the ant be after 2019 moves?
<image1>
Choices:
A. only at $P, R$ or $S$, not at $Q$ or $T$
B. only at $P$, $R$, $S$ or $T$, not at $Q$
C. only at $Q$
D. only at $T$
E. At all of the points
Answer: C
Problem: <image>Question: When Cosme correctly wears his new shirt, as shown on the left figure, the horizontal stripes form seven closed arches around his body. This morning he buttoned his shirt in the wrong way, as shown on the right. How many open arches were there around Cosme's body this morning?
<image1>
Answer: 1
Problem: <image>Question: In the addition beside, different letters represent different numbers and equal letters represent equal numbers. The resulting sum is a number of four digits, B being different from zero. What is the sum of the numbers of this number?
<image1>
Choices:
A. AA
B. BB
C. AB
D. BE
E. EA
Answer: B
Problem: <image>Question: There are several figures that can be formed by nine squares of $1 \mathrm{~cm}$ side by side (see an example beside) and one of them has the biggest perimeter. What is this perimeter?
<image1>
Choices:
A. $12 \mathrm{~cm}$
B. $14 \mathrm{~cm}$
C. $16 \mathrm{~cm}$
D. $18 \mathrm{~cm}$
E. $20 \mathrm{~cm}$
Answer: E
Problem: <image>Question: Four points were marked on a grid of $1 \mathrm{~cm}$ side squares. Of the possible triangular regions that can be obtained with vertices in three of these points, one has the largest area. What is this area, in $\mathrm{cm}^{2}$?
<image1>
Answer: 5.5
Problem: <image>Question: Martinho made a bicolor kite with six pieces of a thin strip of bamboo. Two pieces were used for the diagonals, which are perpendicular. The other four pieces were used to connect the middle points on the sides of the kite, as shown in the picture. What is the ratio between the blue and yellow parts of the kite?
<image1>
Answer: 1
Problem: <image>Question: The shortest way from Atown to Cetown is through Betown. Going back by this road from Cetown to Atown, we first find the signposts on the left side of the road. Further on we find the road signs on the right side of the road. How far is it from Betown to Atown?
<image1>
Choices:
A. $1 \mathrm{~km}$
B. $2 \mathrm{~km}$
C. $3 \mathrm{~km}$
D. $4 \mathrm{~km}$
E. $5 \mathrm{~km}$
Answer: D
Problem: <image>Question: Toninho wants to write strictly positive and consecutive whole numbers, in the nine places of the figure, so that the sum of the three numbers in each diameter is equal to 24. What is the largest possible sum for all the nine numbers?
<image1>
Answer: 81
Problem: <image>Question: Two circles are tangent to each other and also to two sides of a square. What is the measure of the $A \hat{O} B$ angle, determined by three of these points of tangency, as shown in the figure?
<image1>
Choices:
A. $110^{\circ}$
B. $112^{\circ}$
C. $120^{\circ}$
D. $128^{\circ}$
E. $135^{\circ}$
Answer: E
Problem: <image>Question: Ana plays with $n \times n$ boards by placing a token in each of the cells with no common points with other cells containing tokens. In the picture beside we see how to place as many chips as possible on $5 \times 5$ and $6 \times 6$ boards. In this way, how many chips can Ana possibly put on a $2020 \times 2020$ board?
<image1>
Choices:
A. 2020
B. 4039
C. $674^{2}$
D. $1010^{2}$
E. $2020^{2}$
Answer: D
Problem: <image>Question: The next window is a square of area $1 \mathrm{~m}^{2}$ and is composed of four triangles, which areas, indicated in the figure, follow the ratios $3 A=4 B$ and $2 C=3 D$. A fly is placed exactly at the point where these four triangles touch each other. The fly flies directly to the side closest to the window. How much does it fly?
<image1>
Choices:
A. $40 \mathrm{~cm}$
B. $30 \mathrm{~cm}$
C. $25 \mathrm{~cm}$
D. $20 \mathrm{~cm}$
E. $10 \mathrm{~cm}$
Answer: A
Problem: <image>Question: Julia puts the nine chips on the right in a box. She then takes one chip at a time, without looking, and notes down its digit, obtaining, at the end, a number of nine different digits. What is the probability that the number written by Julia is divisible by 45?
<image1>
Choices:
A. $\frac{1}{9}$
B. $\frac{2}{9}$
C. $\frac{1}{3}$
D. $\frac{4}{9}$
E. $\frac{8}{9}$
Answer: A
Problem: <image>Question: A rectangular sheet with one side of $12 \mathrm{~cm}$ is folded along its $20 \mathrm{~cm}$ diagonal. What is the overlapping area of the folded parts, indicated in gray in the picture beside?
<image1>
Choices:
A. $24 \mathrm{~cm}^{2}$
B. $36 \mathrm{~cm}^{2}$
C. $48 \mathrm{~cm}^{2}$
D. $50 \mathrm{~cm}^{2}$
E. $75 \mathrm{~cm}^{2}$
Answer: E
Problem: <image>Question: Julia wrote four positive integers, one at each vertex of a triangular base pyramid. She calculated the sum of the numbers written on the vertices of one face and the product of the numbers written on the vertices of other two faces, obtaining 15, 20 and 30, respectively. What is the highest possible value of the product of the four numbers?
<image1>
Answer: 120
Problem: <image>Question: A large rectangular plot is divided into two lots that are separated from each other by an $A B C D$ fence, as shown in the picture beside. The $A B, B C$ and $C D$ parts of this fence are parallel to the sides of the rectangle and have lengths of $30 \mathrm{~m}$, $24 \mathrm{~m}$ and $10 \mathrm{~m}$, respectively. The owners of these lands have combined to knock down the fence and make a new straight AE fence, without changing the area of each of the lands. How far from point $D$ should the $E$ end of the fence be?
<image1>
Choices:
A. $8 \mathrm{~m}$
B. $10 \mathrm{~m}$
C. $12 \mathrm{~m}$
D. $14 \mathrm{~m}$
E. $16 \mathrm{~m}$
Answer: C
Problem: <image>Question: On the $8 \times 8$ board beside, in how many ways can you place two chips, one green and one red, in different colored cells, so that the chips are not in the same row or in the same column of the board?
<image1>
Answer: 1536
Problem: <image>Question: Jenny looks at her weather app that shows the predicted weather and maximum temperatures for the next five days. Which of the following represents the corresponding graph of maximum temperatures?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: A park is shaped like an equilateral triangle. A cat wants to walk along one of the three indicated paths (thicker lines) from the upper corner to the lower right corner. The lengths of the paths are $P, Q$ and $R$, as shown. Which of the following statements about the lengths of the paths is true?
<image1>
Choices:
A. $P<Q<R$
B. $P<R<Q$
C. $P<Q=R$
D. $P=R<Q$
E. $P=Q=R$
Answer: B
Problem: <image>Question: Six rectangles are arranged as shown. The top left-hand rectangle has height $6 \mathrm{~cm}$. The numbers within the rectangles indicate their areas in $\mathrm{cm}^{2}$. What is the height of the bottom right-hand rectangle?
<image1>
Choices:
A. $4 \mathrm{~cm}$
B. $5 \mathrm{~cm}$
C. $6 \mathrm{~cm}$
D. $7.5 \mathrm{~cm}$
E. $10 \mathrm{~cm}$
Answer: B