Question
A right-angled triangle has sides of length 3 cm, 4 cm and 5 cm. Determine the sizes of all the angles in the triangle, giving your answers to the nearest degree.
Answers
GPT-4o mini
To determine the angles in a right-angled triangle with sides of lengths 3 cm, 4 cm, and 5 cm, we can use the relationships defined by trigonometric functions.
1. Since this is a right-angled triangle, one angle is \(90^\circ\). This angle is opposite the longest side (the hypotenuse) which is 5 cm.
2. To find the other two angles, let's denote the angles opposite the sides of lengths 3 cm and 4 cm as \(A\) and \(B\), respectively.
3. We can use the sine, cosine, or tangent functions to find the angles. Here, we will use the tangent function, which is defined as:
\[
\tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}
\]
Now we can use the inverse tangent function to find angle \(A\):
\[
A = \tan^{-1}\left(\frac{3}{4}\right)
\]
Using a calculator:
\[
A \approx 36.87^\circ
\]
Rounding to the nearest degree, we find that:
\[
A \approx 37^\circ
\]
4. Now, we can find angle \(B\) using the relationship that the sum of the angles in a triangle is \(180^\circ\). Since we know one angle is \(90^\circ\):
\[
B = 90^\circ - A
\]
Thus,
\[
B = 90^\circ - 37^\circ \approx 53^\circ
\]
5. Summarizing the angles, we have:
- Angle \(A \approx 37^\circ\)
- Angle \(B \approx 53^\circ\)
- Angle \(C \approx 90^\circ\) (the right angle)
Therefore, the sizes of all the angles in the triangle, to the nearest degree, are:
- \(A \approx 37^\circ\)
- \(B \approx 53^\circ\)
- \(C = 90^\circ\)
1. Since this is a right-angled triangle, one angle is \(90^\circ\). This angle is opposite the longest side (the hypotenuse) which is 5 cm.
2. To find the other two angles, let's denote the angles opposite the sides of lengths 3 cm and 4 cm as \(A\) and \(B\), respectively.
3. We can use the sine, cosine, or tangent functions to find the angles. Here, we will use the tangent function, which is defined as:
\[
\tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}
\]
Now we can use the inverse tangent function to find angle \(A\):
\[
A = \tan^{-1}\left(\frac{3}{4}\right)
\]
Using a calculator:
\[
A \approx 36.87^\circ
\]
Rounding to the nearest degree, we find that:
\[
A \approx 37^\circ
\]
4. Now, we can find angle \(B\) using the relationship that the sum of the angles in a triangle is \(180^\circ\). Since we know one angle is \(90^\circ\):
\[
B = 90^\circ - A
\]
Thus,
\[
B = 90^\circ - 37^\circ \approx 53^\circ
\]
5. Summarizing the angles, we have:
- Angle \(A \approx 37^\circ\)
- Angle \(B \approx 53^\circ\)
- Angle \(C \approx 90^\circ\) (the right angle)
Therefore, the sizes of all the angles in the triangle, to the nearest degree, are:
- \(A \approx 37^\circ\)
- \(B \approx 53^\circ\)
- \(C = 90^\circ\)