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∆ABC has side lengths of 10 units, 20 units, and 24 units. ∆XYZ is similar to ∆ABC, and the length of its longest side is 60 units. The perimeter of ∆XYZ is units. If the height of ∆ABC, with respect to its longest side being the base, is 8 units, the area of ∆XYZ is square units.

Respuesta :

ImKC

Answer: Perimeter = 135 units      Area = 1200 Square units

Step-by-step explanation: I think you want the perimeter and area of XYZ so that's what I will answer for.

First, we are given that ABC and XYZ are proportional and that their longest sides are 24 units and 60 units respectively.

Using this, we can say XYZ = ABC * 5/2 (60/24 = 5/2)

Therefore, the other two sides of XYZ are 50 and 25.

We can get the perimeter using 60 + 50 + 25, that equals 135 units

Next, since the height of ABC was 8 units with 24 units being its base, it's likely safe to say the height = 1/3 base

We will apply with to triangle XYZ, height = 1/3 * 60 = 20

20 units * 60 units = 1200 square units

The perimeter and area of the ΔXYZ are 135 units and 600 sq. units. Where similar triangles are related by a scale factor.

How to find the scale factor for similar triangles?

The ratio of their respective sides of two similar triangles gives the scale factor. I.e.,

Consider ΔABC and ΔDEF are two similar triangles

Then its scale factor = DE/AB = EF/BC = DF/AC

The scale factor is also defined as follows:

(Scale factor)² = (Area of the triangle DEF)/(Area of the triangle ABC)

or

Scale factor = (perimeter of ΔDEF)/(perimeter of ΔABC)

Finding the scale factor:

Given that the sides of the triangle ABC are 10 units, 20 units, and 24 units. The longest side is 24 units.

The triangle XYZ has the longest side of length 60 units.

Since ΔABC ~ ΔXYZ

So, the ratio of their longest sides gives the scale factor. I.e.,

Scale factor = 60/24 = 2.5

Calculating the perimeter of the ΔXYZ:

The perimeter of the ΔXYZ is calculated by

Scale factor = (perimeter of the ΔXYZ)/(perimeter of the ΔABC)

⇒ 2.5 = (perimeter of the ΔXYZ)/(10 + 20 + 24)

⇒ perimeter of the ΔXYZ = 2.5 × 54

∴ the perimeter of the ΔXYZ = 135 units

Calculating the area of the ΔXYZ:

The area of the ΔXYZ is calculated by

(Scale factor)² = (area of the ΔXYZ)/(area of the ΔABC)

So, the area of the ΔABC, whose height h = 8 units and base b = 24 units is

Area of the ΔABC = 1/2 × b × h

                              = 1/2 × 24 × 8

                              = 96 sq. units

Thus,

(Scale factor)² = (area of the ΔXYZ)/(area of the ΔABC)

⇒ (2.5)² =  (area of the ΔXYZ)/(96)

⇒  area of the ΔXYZ = (2.5)² × 96

∴  area of the ΔXYZ = 600 sq units

Thus, the perimeter and area of the ΔXYZ are 135 units and 600 sq. units.

Learn more about the scale factor of similar triangles here:

https://brainly.com/question/1555859

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