Right Angled Triangle Side Length Calculation
The problem involves finding the length of a side in a right-angled triangle using the Pythagorean theorem.
Applying the Pythagorean Theorem
In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs). The triangle has a right angle at \(A\), making \(BC\) the hypotenuse.
The theorem is stated as:
\(AB^2 + AC^2 = BC^2\)
Given Values
- Length of side \(AB = 3\) cm
- Length of hypotenuse \(BC = 12\) cm
- Length of side \(AC = ?\)
Calculation Steps
- Substitute the known values into the Pythagorean theorem:
\(3^2 + AC^2 = 12^2\)
- Calculate the squares:
\(9 + AC^2 = 144\)
- Isolate \(AC^2\) by subtracting 9 from both sides:
\(AC^2 = 144 - 9\)
\(AC^2 = 135\)
- Find the length of \(AC\) by taking the square root of 135:
\(AC = \sqrt{135}\)
- Simplify the radical \(\sqrt{135}\):
Find the prime factors of 135: \(135 = 5 \times 27 = 5 \times 3^3 = 3 \times 3^2 \times 5 = 9 \times 15\).
\(AC = \sqrt{9 \times 15}\)
\(AC = \sqrt{9} \times \sqrt{15}\)
\(AC = 3\sqrt{15} \text{ cm}\)
Conclusion
The length of side \(AC\) is \(3\sqrt{15}\) 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 \(QR\)
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 $AC$?
In a right angled triangle, with angle at A being , the side $AB$ is of length 4cm and $BC$ is 15 cm. What is the length of side $AC$?
- Triangles $PQR$ and $ABC$ are similar triangles.
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 $RQ$ =? - 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?
- In a triangle the length of the sides, $AB$ is 3m and $BC$ is 5m what is the length of $AC$, if the angle formed at $B$ is $60^\circ$
- 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 \(QR\)
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 $AC$?
In a right angled triangle, with angle at A being , the side $AB$ is of length 4cm and $BC$ is 15 cm. What is the length of side $AC$?
