Question

Triangle ABC is mathematically similar to triangle PQR.The area of triangle ABC is 16 . 2

a. Calculate the area of triangle PQR

b. The triangles are the cross-sections of prisms which are
also mathematically similar.The volume of the smaller prism
is 320 .Calculate the length of the larger prism

tolong ya besok Lusa dikumpulkan

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no ngawur
pakai cara ya yang lengkap

aku janji banget buat gak nyontek

​

Answer 1

Verified answer

Jawaban:

(1) ANSWER NUMBER 1 !----------------------------------------------

Sure, here are the answers to the questions:

a. Calculate the area of triangle PQR

Since triangles ABC and PQR are mathematically similar, their areas are proportional to the squares of their corresponding side lengths. Therefore, we can set up the following proportion:

(Area of triangle ABC) / (Area of triangle PQR) = (side length of triangle ABC)^2 / (side length of triangle PQR)^2

We are given that the area of triangle ABC is 16 cm^2. Let x be the side length of triangle PQR. Substituting the given values into the proportion, we get:

16 / x^2 = 1^2 / x^2

Simplifying, we get:

16 = 1

This is clearly a contradiction, so there must be an error in the given information or the question itself.

b. Calculate the length of the larger prism

Since the triangles are the cross-sections of prisms which are also mathematically similar, their volumes are proportional to the cubes of their corresponding side lengths. Therefore, we can set up the following proportion:

(Volume of the smaller prism) / (Volume of the larger prism) = (side length of the smaller prism)^3 / (side length of the larger prism)^3

We are given that the volume of the smaller prism is 320 cm^3. Let Y be the side length of the larger prism. Substituting the given values into the proportion, we get:

320 / Y^3 = 1^3 / Y^3

(2) ANSWER NUMBER 2 !---------------------------------------------

a. Calculate the area of triangle PQR

Since triangles ABC and PQR are mathematically similar, their corresponding sides are proportional. Let the ratio of corresponding sides be k, so we have:

AB/PQ = AC/PR = BC/QR = k

We know that the area of a triangle is given by:

Area = 0.5 * base * height

Let's denote the area of triangle PQR as Area_PQR, the base of triangle PQR as Base_PQR, and the height of triangle PQR as Height_PQR. We can express the area of triangle PQR as:

Area_PQR = 0.5 * Base_PQR * Height_PQR

We are given that the area of triangle ABC is 16 cm², so we can write:

Area_ABC = 0.5 * Base_ABC * Height_ABC = 16 cm²

Since triangles ABC and PQR are similar, we can express the area of triangle PQR in terms of the corresponding sides of triangle ABC:

Area_PQR = k² * Area_ABC

Substituting the given value of Area_ABC, we get:

Area_PQR = k² * 16 cm²

We need to find the value of k to calculate the area of triangle PQR.

b. Calculate the length of the larger prism

The volume of a prism is given by:

Volume = Base Area * Height

Let's denote the volume of the smaller prism as Volume_S, the base area of the smaller prism as Base_S, the height of the smaller prism as Height_S, the volume of the larger prism as Volume_L, the base area of the larger prism as Base_L, and the height of the larger prism as Height_L. We are given that the volume of the smaller prism is 320 cm³, so we can write:

Volume_S = Base_S * Height_S = 320 cm³

Since the prisms are mathematically similar, their corresponding dimensions are proportional. Let the ratio of corresponding dimensions be k, so we have:

Base_L/Base_S = Height_L/Height_S = k

We can express the volume of the larger prism in terms of the corresponding dimensions of the smaller prism:

Volume_L = k³ * Volume_S

Substituting the given value of Volume_S, we get:

Volume_L = k³ * 320 cm³

We need to find the value of k to calculate the length of the larger prism.

To solve for k, we can use the fact that the triangles ABC and PQR are similar. We know that the area of triangle ABC is 16 cm², so we can write:

Area_ABC = 0.5 * Base_ABC * Height_ABC = 16 cm²

We can assume that the base and height of triangle ABC are represented by variables, let's say Base_ABC and Height_ABC. Substituting these variables, we get:

0.5 * Base_ABC * Height_ABC = 16 cm²

Since the triangles are similar, we can express the corresponding sides of triangle PQR in terms of the corresponding sides of triangle ABC:

Base_PQR = k * Base_ABC

Height_PQR = k * Height_ABC

We can express the area of triangle PQR in terms of the corresponding sides of triangle PQR:

Area_PQR = 0.5 * Base_PQR * Height_PQR

Substituting the expressions for Base_PQR and Height_PQR, we get:

Area_PQR = 0.5 * k * Base_ABC * k * Height_ABC

Since the triangles are similar, we can set the areas equal to each other:

0.5 * Base_ABC * Height_ABC = 0.5 * k * Base_ABC * k * Height_ABC

Canceling out the common factors, we get:

1 = k²

Therefore, k = 1.

Now that we know k, we can calculate the area of triangle PQR and the length of the larger prism.

a. Area of triangle PQR

Using the value of k, we can express the area of triangle PQR as:

Area_PQR = k² * Area_ABC = 1² * 16 cm² = 16 cm²

Therefore, the area of triangle PQR is 16 cm².

b. Length of the larger prism

Using the value of k, we can express the volume of the larger prism as:

Volume_L = k³ * Volume_S = 1³ * 320 cm³ = 320 cm³

We can express the height

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