Triangle Classification Using Side Lengths
We are given a triangle \(\Delta\text{ABC}\) with the following side lengths:
- \(\text{AB} = 16\text{cm}\)
- \(\text{BC} = 63\text{cm}\)
- \(\text{AC} = 65\text{cm}\)
Pythagorean Theorem Converse Check
To determine if the triangle is right-angled, acute, or obtuse, we use the converse of the Pythagorean theorem. We need to compare the square of the longest side with the sum of the squares of the other two sides.
Calculating Side Squares
First, calculate the squares of each side length:
- Square of side AB: \(\text{AB}^2 = (16\text{cm})^2 = 256\text{cm}^2\)
- Square of side BC: \(\text{BC}^2 = (63\text{cm})^2 = 3969\text{cm}^2\)
- Square of side AC (the longest side): \(\text{AC}^2 = (65\text{cm})^2 = 4225\text{cm}^2\)
Verifying the Theorem Condition
Next, sum the squares of the two shorter sides:
\(\text{AB}^2 + \text{BC}^2 = 256\text{cm}^2 + 3969\text{cm}^2 = 4225\text{cm}^2\)
Compare this sum to the square of the longest side:
We found that \(\text{AB}^2 + \text{BC}^2 = 4225\text{cm}^2\), and \(\text{AC}^2 = 4225\text{cm}^2\). Therefore, \(\text{AB}^2 + \text{BC}^2 = \text{AC}^2\).
Conclusion on Triangle Type
According to the converse of the Pythagorean theorem, if \(a^2 + b^2 = c^2\), the triangle is a right-angled triangle. Since the condition is met, triangle \(\Delta\text{ABC}\) is a Right angled Triangle.
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 $AC$?
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
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 $AC$?
