Sure! To find the three angles of a right-angled triangle where the smallest angle and the largest angle are in the ratio 2:5, we can follow these steps:
1. Let's denote the smallest angle as [tex]\( A \)[/tex] and the largest angle (other than the right angle) as [tex]\( B \)[/tex].
2. Since it's a right-angled triangle, one of the angles is [tex]\( 90^\circ \)[/tex]. Let’s denote this angle as [tex]\( C \)[/tex].
Given that the smallest angle and the largest angle are in the ratio 2:5, we can express [tex]\( A \)[/tex] and [tex]\( B \)[/tex] in terms of a common variable [tex]\( x \)[/tex]:
[tex]\[ A = 2x \][/tex]
[tex]\[ B = 5x \][/tex]
3. In any triangle, the sum of the angles must be [tex]\( 180^\circ \)[/tex]. Therefore, we have:
[tex]\[ A + B + C = 180^\circ \][/tex]
Since [tex]\( C \)[/tex] is [tex]\( 90^\circ \)[/tex]:
[tex]\[ 2x + 5x + 90^\circ = 180^\circ \][/tex]
4. Simplify the equation:
[tex]\[ 7x + 90^\circ = 180^\circ \][/tex]
5. Subtract [tex]\( 90^\circ \)[/tex] from both sides:
[tex]\[ 7x = 90^\circ \][/tex]
6. Divide by 7 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{90}{7} \approx 12.857 \][/tex]
7. Now compute the values of the angles [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A = 2x = 2 \times 12.857 \approx 25.714^\circ \][/tex]
[tex]\[ B = 5x = 5 \times 12.857 \approx 64.286^\circ \][/tex]
So, the three angles of the triangle are:
[tex]\[ \boxed{25.714^\circ, 90^\circ, 64.286^\circ} \][/tex]
