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.
Similar Questions
In the given figure, if \(RS = 3\sqrt{3} \text{ cm}\) and \(RPS\) is an equilateral triangle, then find the value of \(QR\)
- In a right angled triangle, with angle at \(A\) being \(90^\circ\), the side \(AB\) is of length 3cm and \(BC\) is 12 cm. What is the length of side \(AC\)?
For a pair of similar triangles shown in the figure, the angles made at $B$ and $R$ are same in both the triangles $BAC$ and $RAQ$. If the ratio of the areas of the two triangles is 2:1, and if the area of $BAC$ is $100\text{ cm}^2$ then what is the area of $RAQ$ in sq.cm.?
- In a triangle $ABC$, the length of the side $AB$ is 2m, the angle at $C$ is $60^\circ$ and the angle at $B$ is $30^\circ$, what is the length of side $AC$?
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- What is the area of the triangle in sq. cm. with a base of 10 cm and sides of length 6 cm each?
What is the angle at $B$ in the triangle $ABC$, if the angle made at $K$ is $30^\circ$ and the triangles shown in the figure are similar?
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- In a triangle \(\Delta ABC\ AB= 13\text{ cm}, BC=84\text{cm}, \text{and } \angle B=90^\circ\), then length of AC will be
- Which of the following is not a Pythagorean triplet?
Important Questions from Triangles
What is \(AB + BC\) equal to?
In the given figure, if \(RS = 3\sqrt{3} \text{ cm}\) and \(RPS\) is an equilateral triangle, then find the value of \(QR\)
- In a right angled triangle, with angle at \(A\) being \(90^\circ\), the side \(AB\) is of length 3cm and \(BC\) is 12 cm. What is the length of side \(AC\)?
For a pair of similar triangles shown in the figure, the angles made at $B$ and $R$ are same in both the triangles $BAC$ and $RAQ$. If the ratio of the areas of the two triangles is 2:1, and if the area of $BAC$ is $100\text{ cm}^2$ then what is the area of $RAQ$ in sq.cm.?
- In a triangle $ABC$, the length of the side $AB$ is 2m, the angle at $C$ is $60^\circ$ and the angle at $B$ is $30^\circ$, what is the length of side $AC$?
