Triangle $ABC$ have the following sides.
$AB = 36\text{ cm}$
$AC = 20\text{ cm}$
$BC = 48\text{ cm}$
Triangle $PQR$ have the following sides:
$PQ = 9\text{ cm}$
$PR = 5\text{ cm}$
What is the length of side $RQ$ =?
Similar Triangles PQR and ABC: Finding Side RQ
The problem states that triangle $PQR$ is similar to triangle $ABC$ ($\triangle PQR \sim \triangle ABC$). This similarity implies that the ratios of their corresponding sides are equal.
Corresponding Sides Identification
Given the similarity $\triangle PQR \sim \triangle ABC$, the corresponding sides are:
- $PQ$ corresponds to $AB$
- $PR$ corresponds to $AC$
- $RQ$ corresponds to $BC$
Setting Up Proportions
The property of similar triangles allows us to write the following proportion:
$ \frac{PQ}{AB} = \frac{PR}{AC} = \frac{RQ}{BC} $Calculating the Scale Factor
We are given the lengths:
- $AB = 36$ cm, $AC = 20$ cm, $BC = 48$ cm
- $PQ = 9$ cm, $PR = 5$ cm
We can find the scale factor using the pairs $(PQ, AB)$ or $(PR, AC)$.
Using $PQ$ and $AB$: $ \text{Scale Factor} = \frac{PQ}{AB} = \frac{9 \text{ cm}}{36 \text{ cm}} = \frac{1}{4} $
Using $PR$ and $AC$: $ \text{Scale Factor} = \frac{PR}{AC} = \frac{5 \text{ cm}}{20 \text{ cm}} = \frac{1}{4} $
The scale factor from $\triangle ABC$ to $\triangle PQR$ is consistent and equals $\frac{1}{4}$.
Determining the Length of RQ
Now, we use the scale factor to find the length of $RQ$. We set up the proportion involving $RQ$ and its corresponding side $BC$:
$ \frac{RQ}{BC} = \text{Scale Factor} $ $ \frac{RQ}{48 \text{ cm}} = \frac{1}{4} $To find $RQ$, multiply both sides by $48$ cm:
$ RQ = \frac{1}{4} \times 48 \text{ cm} $ $ RQ = 12 \text{ cm} $Therefore, the length of side $RQ$ is 12 cm.
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