Can you now compare the lengths and ?

We can compare the lengths by using the properties of a right-angled triangle shown in the diagram. Step 1 — Identify the right-angled triangle Let us look at the triangle formed by points A, B, and E. This is triangle ABE. The diagram shows a small square symbol at point E. This symbol means that the angle at E, , is . So, triangle ABE is a right-angled triangle. Step 2 — Identify the sides of the triangle In any right-angled triangle, the side opposite the angle is called the hypotenuse. In triangle ABE, the side opposite is AB. So, AB is the hypotenuse. Its length is given as . The other two sides, AE and BE, are called the legs of the triangle. The length of AE is given as . Step 3 — Compare the lengths using the Pythagorean theorem The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, for triangle ABE: We are given that and . From the diagram, point B is clearly not the same as point E (B is off the plane, E is on the line AE). This means the length of BE must be greater than 0. So, is a positive number. Since is equal to plus a positive number, must be greater than . Since and represent lengths, they must be positive values. If and both and are positive, then must be greater than . > Answer The length is greater than the length .

Fractals and Visualising Solids | IT

Question 30

Can you now compare the lengths $p$ and $l$ ?

Solution

We can compare the lengths by using the properties of a right-angled triangle shown in the diagram.

Step 1 — Identify the right-angled triangle

Let us look at the triangle formed by points A, B, and E. This is triangle ABE. The diagram shows a small square symbol at point E. This symbol means that the angle at E, $\angle AEB$ , is $90^\circ$ . So, triangle ABE is a right-angled triangle.

Step 2 — Identify the sides of the triangle

In any right-angled triangle, the side opposite the $90^\circ$ angle is called the hypotenuse. In triangle ABE, the side opposite $\angle AEB$ is AB. So, AB is the hypotenuse. Its length is given as $l$ . The other two sides, AE and BE, are called the legs of the triangle. The length of AE is given as $p$ .

Step 3 — Compare the lengths using the Pythagorean theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, for triangle ABE: $AB^2 = AE^2 + BE^2$ We are given that $AB = l$ and $AE = p$ . $l^2 = p^2 + BE^2$ From the diagram, point B is clearly not the same as point E (B is off the plane, E is on the line AE). This means the length of BE must be greater than 0. So, $BE^2$ is a positive number. Since $l^2$ is equal to $p^2$ plus a positive number, $l^2$ must be greater than $p^2$ . $l^2 > p^2$ Since $l$ and $p$ represent lengths, they must be positive values. If $l^2 > p^2$ and both $l$ and $p$ are positive, then $l$ must be greater than $p$ . $l > p$

$\boxed{l > p}$

Diagram 1

Answer

The length $l$ is greater than the length $p$ .

More questions in IT

Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Carpet.

By its construction, each step in the sequence has (i) squares of the same size that remain in the figure, and the size of these squares becomes smaller and smaller as the step number increases, and (ii) square holes that are formed by removing square pieces.

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[Hint: Note that the corner triangles are isosceles.]

This fractal is called the Sierpinski Triangle/Gasket.

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Can you describe a solid and a viewpoint that would result in each of the following cases? If it helps, you can imagine the solid passing through a wall like Tom did, and leaving a hole of the appropriate shape.

A solid whose profile has a square outline
A solid whose profile has a circular outline
A solid whose profile has a triangular outline

As we saw with the elephant, a given solid might have very different profiles from different viewpoints. Can you visualise solids that have the following contrasting profiles?

Spend some time on this, and if you are finding it difficult to visualise, you may look around and use objects that are around you, or that you will make in the next section. Feel free to consider viewpoints from any direction, including directly above the object.

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Visualise how it can be folded to form a cube.

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Q11

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Q12

Draw a net with appropriate measurements that can be folded into a regular tetrahedron. Verify if it works by making an actual cutout.

Q13

Draw a net with appropriate measurements that can be folded into a square pyramid. Verify if it works by making an actual cutout.

Q14

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If the circular faces of a cylinder are unfolded, and if a cut is made along the height of the cylinder, as shown in the figure below, then we get

Q15

What are the sidelengths of the rectangle obtained?

Q16

How will the net of a cone look?

Q17

If the cone is slit open along the line $l$ and then unrolled, what will we get?

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Q18

What surface do you construct by using the above net, in which O is not the centre of the boundary circle? Make a physical model to help you answer this question!

Q19

Draw a net with appropriate measurements that can be folded into a triangular prism. Verify that it works by making an actual cutout.

Q20

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Q21

What is the shortest path for the ant to reach the laddu?

Q22

What about in the following case?

Q23

If we think that a certain path is the shortest, how can we be sure that it truly is, among all the infinite possibilities?

Q24

For example, are either of these the shortest path?

Q25

What does this show?

Q26

Have we now completely analysed the problem of finding the shortest path between two points on a cuboid?

Q27

What is the length of the shortest path between the ant and the laddu?

Q29

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Q30

Can you now compare the lengths $p$ and $l$ ?

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Q32

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Q34

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Q35

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Q36

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Q40

Context: In Fig. 4.8, there are five basic shapes in Tetris, corresponding to the different ways of arranging four squares.

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Q41

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Q42

Can you try drawing the other tetris shapes on isometric paper?