Math / Geometry

Question0. If $\triangle K L M \sim \triangle P Q R$ with a scale factor of 3:5, find the perimeter of $\triangle P Q R$ .

Studdy Solution

STEP 1

What is this asking? If we have two similar triangles and we know the lengths of the sides of one and the ratio of their sizes, what's the total length around the bigger triangle? Watch out! Don't mix up the scale factor!
It's easy to flip it around accidentally.

STEP 2

1. Find the perimeter of the smaller triangle.
2. Calculate the perimeter of the larger triangle.

STEP 3

Alright, let's kick things off with the smaller triangle, $\triangle KLM$ !
We've got the lengths of its sides: $6$ , $12$ , and $15$ .
The perimeter is just the sum of all the sides.

STEP 4

So, let's add them up: $6 + 12 + 15 = 33$ .
The perimeter of $\triangle KLM$ is $33$ units!

STEP 5

Now, for the star of the show, $\triangle PQR$ !
We know $\triangle KLM$ and $\triangle PQR$ are similar with a scale factor of $3:5$ .
This means if a side of $\triangle KLM$ is $3$ units long, the corresponding side of $\triangle PQR$ is $5$ units long.

STEP 6

We can write this as a ratio: $\frac{\text{Perimeter of } \triangle PQR}{\text{Perimeter of } \triangle KLM} = \frac{5}{3}$

STEP 7

We already know the perimeter of $\triangle KLM$ is $33$ , so let's plug that in: $\frac{\text{Perimeter of } \triangle PQR}{33} = \frac{5}{3}$

STEP 8

To isolate the "Perimeter of $\triangle PQR$ ", we can multiply both sides of the equation by $33$ : $\text{Perimeter of } \triangle PQR = \frac{5}{3} \cdot 33$

STEP 9

Let's simplify!
We can think of $33$ as $3 \cdot 11$ , so: $\text{Perimeter of } \triangle PQR = \frac{5}{3} \cdot (3 \cdot 11)$ $\text{Perimeter of } \triangle PQR = \frac{5 \cdot 3 \cdot 11}{3}$ Since we're multiplying and dividing by 3, we can divide to one: $\text{Perimeter of } \triangle PQR = 5 \cdot 11$ $\text{Perimeter of } \triangle PQR = 55$

STEP 10

The perimeter of $\triangle PQR$ is $\textbf{55}$ units!

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