Question
solve the following problems
Given:Triangle AKL,AK=9,m<K=90 degrees, m<A=60 degrees.
Find:The perimeter of the triangle.
Given:Triangle AKL,AK=9,m<K=90 degrees, m<A=60 degrees.
Find:The perimeter of the triangle.
Answers
GPT-4o mini
To find the perimeter of triangle \( AKL \), we first need to determine the lengths of sides \( AL \) and \( KL \).
Given:
- \( AK = 9 \)
- \( m \angle K = 90^\circ \)
- \( m \angle A = 60^\circ \)
This implies that triangle \( AKL \) is a right triangle at \( K \). The angles of triangle \( AKL \) add up to \( 180^\circ \), so we can find the measure of angle \( L \):
\[
m \angle L = 180^\circ - m \angle K - m \angle A = 180^\circ - 90^\circ - 60^\circ = 30^\circ
\]
Thus, we have the angles:
- \( m \angle A = 60^\circ \)
- \( m \angle K = 90^\circ \)
- \( m \angle L = 30^\circ \)
Now we can use the relationships in a 30-60-90 triangle to find the lengths of sides \( AL \) and \( KL \).
In a 30-60-90 triangle, the ratios of the lengths of the sides opposite the angles are as follows:
- The side opposite the \( 30^\circ \) angle is \( x \).
- The side opposite the \( 60^\circ \) angle is \( x\sqrt{3} \).
- The side opposite the \( 90^\circ \) angle (the hypotenuse) is \( 2x \).
In triangle \( AKL \):
- \( AK \) serves as the hypotenuse, corresponding to \( 90^\circ \) and has a length of \( 9 \).
- Therefore, \( 2x = 9 \) implies \( x = \frac{9}{2} = 4.5 \).
Using \( x \):
- The length \( AL \), opposite the \( 60^\circ \) angle, is \( x\sqrt{3} = 4.5\sqrt{3} \).
- The length \( KL \), opposite the \( 30^\circ \) angle, is \( x = 4.5 \).
Now we calculate these lengths numerically:
\[
AL = 4.5\sqrt{3} \approx 4.5 \times 1.732 \approx 7.794
\]
\[
KL = 4.5
\]
Finally, we find the perimeter \( P \) of triangle \( AKL \):
\[
P = AK + AL + KL = 9 + 4.5\sqrt{3} + 4.5 \approx 9 + 7.794 + 4.5 \approx 21.294
\]
So the perimeter of triangle \( AKL \) is approximately \( \boxed{21.294} \).
Given:
- \( AK = 9 \)
- \( m \angle K = 90^\circ \)
- \( m \angle A = 60^\circ \)
This implies that triangle \( AKL \) is a right triangle at \( K \). The angles of triangle \( AKL \) add up to \( 180^\circ \), so we can find the measure of angle \( L \):
\[
m \angle L = 180^\circ - m \angle K - m \angle A = 180^\circ - 90^\circ - 60^\circ = 30^\circ
\]
Thus, we have the angles:
- \( m \angle A = 60^\circ \)
- \( m \angle K = 90^\circ \)
- \( m \angle L = 30^\circ \)
Now we can use the relationships in a 30-60-90 triangle to find the lengths of sides \( AL \) and \( KL \).
In a 30-60-90 triangle, the ratios of the lengths of the sides opposite the angles are as follows:
- The side opposite the \( 30^\circ \) angle is \( x \).
- The side opposite the \( 60^\circ \) angle is \( x\sqrt{3} \).
- The side opposite the \( 90^\circ \) angle (the hypotenuse) is \( 2x \).
In triangle \( AKL \):
- \( AK \) serves as the hypotenuse, corresponding to \( 90^\circ \) and has a length of \( 9 \).
- Therefore, \( 2x = 9 \) implies \( x = \frac{9}{2} = 4.5 \).
Using \( x \):
- The length \( AL \), opposite the \( 60^\circ \) angle, is \( x\sqrt{3} = 4.5\sqrt{3} \).
- The length \( KL \), opposite the \( 30^\circ \) angle, is \( x = 4.5 \).
Now we calculate these lengths numerically:
\[
AL = 4.5\sqrt{3} \approx 4.5 \times 1.732 \approx 7.794
\]
\[
KL = 4.5
\]
Finally, we find the perimeter \( P \) of triangle \( AKL \):
\[
P = AK + AL + KL = 9 + 4.5\sqrt{3} + 4.5 \approx 9 + 7.794 + 4.5 \approx 21.294
\]
So the perimeter of triangle \( AKL \) is approximately \( \boxed{21.294} \).