QuestionJune 6, 2025

16. (Trig or special right triangles) OPEN RESPONSE A 10-foot-long ladder leans against a wall so that the ladder and the wall form a 30^circ angle. What is the distance to the nearest tenth of a foot from the ground to the point where the ladder touches the wall?

16. (Trig or special right triangles) OPEN RESPONSE A 10-foot-long ladder leans against a wall so that the ladder and the wall form a 30^circ angle. What is the distance to the nearest tenth of a foot from the ground to the point where the ladder touches the wall?
16. (Trig or special right triangles) OPEN RESPONSE A 10-foot-long ladder leans against a wall so that the ladder and the wall form a 30^circ angle. What is the distance to the nearest tenth of a foot from the ground to the point where the ladder touches the wall?

Solution
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Answer

8.7 feet Explanation 1. Identify the triangle type The ladder forms a 30^{\circ} angle with the wall, creating a 30-60-90 special right triangle. 2. Apply the 30-60-90 triangle ratio In a 30-60-90 triangle, the ratio of the sides opposite the 30^{\circ}, 60^{\circ}, and 90^{\circ} angles are 1:\sqrt{3}:2. The hypotenuse is twice the length of the side opposite the 30^{\circ} angle. 3. Calculate the height The hypotenuse (ladder) is 10 feet. The side opposite the 60^{\circ} angle (height from ground to wall) is \frac{\sqrt{3}}{2} times the hypotenuse. So, height = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3}. 4. Approximate the height Calculate 5\sqrt{3} \approx 5 \times 1.732 = 8.66.

Explanation

1. Identify the triangle type<br /> The ladder forms a $30^{\circ}$ angle with the wall, creating a $30-60-90$ special right triangle.<br /><br />2. Apply the 30-60-90 triangle ratio<br /> In a $30-60-90$ triangle, the ratio of the sides opposite the $30^{\circ}$, $60^{\circ}$, and $90^{\circ}$ angles are $1:\sqrt{3}:2$. The hypotenuse is twice the length of the side opposite the $30^{\circ}$ angle.<br /><br />3. Calculate the height<br /> The hypotenuse (ladder) is 10 feet. The side opposite the $60^{\circ}$ angle (height from ground to wall) is $\frac{\sqrt{3}}{2}$ times the hypotenuse. So, height = $10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3}$.<br /><br />4. Approximate the height<br /> Calculate $5\sqrt{3} \approx 5 \times 1.732 = 8.66$.
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