Question

С точки, которая находится на расстоянии 12 см от плоскости, проведено к этой плоскости две наклонные, угол между которыми 90°. Найдите угол между проекциями наклонных, которые равны 9 см и 16 см. Помогите, если можно с рисунком

Answer 1

МО⊥пл.α   ⇒  МО⊥АО  и  МО⊥ВО
ΔАОМ и ΔВОМ - прямоугольные
АМ²=АО²+МО²=12²+9²=225=15²  ,  АМ=15
ВМ²=ВО²+ВМ²=16²+12²=400=20²  ,  ВМ=20
∠АМВ=90°  по условию  ⇒  АВ²=ВМ²+АМ²=15²+20²=625=25² , АВ=25
АО+ВО=9+16=25=АВ  ⇒  ∠АОВ=180°
Если не заметить, что АО+ВО=АВ, то чтобы найти ∠АОВ, надо воспользоваться теоремой косинусов в ΔАОВ:
25²=9²+16²-2·9·16·сosα
625=337-288·cosα
288=-288·cosα
cosα=-1  ⇒  α=180°

Answer 2

Calculating the Angle Between Projections of Inclined Lines

To find the angle between the projections of the inclined lines, we can use the given information about the distances and angles. Let's calculate the angle step by step.

1. Given Information: - Distance from the point to the plane: 12 cm - Two inclined lines with an angle of 90° between them - Projections of the lines are 9 cm and 16 cm

2. Calculating the Angle: - First, we need to find the lengths of the inclined lines. Using the given distance of 12 cm and the projections of 9 cm and 16 cm, we can apply trigonometric ratios to find the lengths of the lines.

- Using the Pythagorean theorem, we can calculate the lengths of the inclined lines: - For the line with the 9 cm projection: - Let the length of this line be x. - Using the distance (12 cm) and the projection (9 cm), we can use the Pythagorean theorem: \(x^2 = 12^2 - 9^2\) - Solving for x, we get the length of the line.

- For the line with the 16 cm projection: - Let the length of this line be y. - Using the distance (12 cm) and the projection (16 cm), we can use the Pythagorean theorem: \(y^2 = 12^2 - 16^2\) - Solving for y, we get the length of the line.

- Once we have the lengths of the inclined lines, we can use trigonometric ratios to find the angle between them.

3. Conclusion: - After calculating the lengths of the inclined lines, we can use the lengths to find the angle between the lines.

Let's proceed with the calculations and find the angle between the projections of the inclined lines.