Question

The areas of two similar triangles are 42cm and 262.5cm. if the altitude of the smaller triangle is 7cm. what is the length of the base of the larger triangle?

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Answer 1

To find the length of the base of the larger triangle, you can use the concept of similar triangles and the proportion of their corresponding sides. By setting up and solving a proportion, you can find the length of the base of the larger triangle.

Step-by-step explanation:

To find the length of the base of the larger triangle, we need to use the concept of similar triangles and the proportion of their corresponding sides. Let's assume the lengths of the bases of the smaller and larger triangles are 'b' and 'B' respectively. Since the areas of the triangles are proportional to the squares of their corresponding sides, we can set up the following proportion:

(b^2)/(B^2) = (42)/(262.5)

Simplifying the proportion, we get (b^2)/(B^2) = (2/15).

Taking the square root of both sides gives: b/B = sqrt(2/15).

Since we know that the altitude of the smaller triangle is 7cm, we can use this information to find the length of the base of the smaller triangle using the area formula of a triangle:

(1/2) * b * 7 = 42cm

Solving this equation gives b = 12cm.

Substituting the value of b in the proportion equation, we have 12/B = sqrt(2/15).

Cross multiplying and solving for B, we get B = 12/sqrt(2/15).

Calculating this value gives B ≈ 26.67cm.

Answer 2

75 cm

Step-by-step explanation:

It is given that the triangles are similar. It means that the ratio of their areas are equal to the ratio of their bases and ratio of their altitudes.

Let
`A_1` be the area of 1st triangle.

Let
`A_2` be the area of 2nd triangle.

Let
`h_1` be the altitude of 1st triangle.

Let
`h_2` be the altitude of 2nd triangle.

Let
`b_1` be the base of 1st triangle.

Let
`b_2` be the base of 2nd triangle.

Then
`A_1: A_2 = h_1 : h_2 = b_1 : b_2 ...... (1)`

`A_1 = 42 cm^(2) \\A_2 = 262.5 cm^(2)\\h_1 = 7 cm\\b_2 = ?`

We know that area of a triangle is:

`A = (1)/(2) * b * h`

Area of smaller triangle:

`(1)/(2) * b_1 * 7 = 42\\\Rightarrow b_1 = 12 cm`

Now, using part of equation (1):

`A_1: A_2 = b_1 : b_2 \\\Rightarrow (42)/(262.5) = (12)/(b_2)\\\Rightarrow b_2 = 75 cm`

Hence, base of larger triangle = 75 cm