Height and Distances

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In a triangle, what is the relationship between the internal bisector of an angle and the opposite side?

The internal bisector is perpendicular to the opposite side.

The internal bisector divides the opposite side into two equal parts.

The internal bisector divides the opposite side in the ratio of the exterior angles.

The internal bisector divides the opposite side in the ratio of the arms of the angle. (correct)

What is the name of the formula used to find the length of a side in a triangle, other than the right-angled triangle?

Tangent formula

Sine formula

Pythagorean formula

Cosine formula (correct)

What is the relationship between the corresponding sides of similar triangles?

They are perpendicular to each other.

They are equal in length.

They are proportional. (correct)

They are parallel to each other.

What is the bearing of an object P from an observer O, if the observer and the object are on the same level?

The angle between the line of sight and the East direction. Signup and view all the answers

What is the exterior angle of a triangle equal to?

The sum of the interior opposite angles. Signup and view all the answers

What is the formula for finding the height of a tower, where H = ?, in terms of angles α and β?

cot β - cot α Signup and view all the answers

In a right-angled triangle, what is the relation between angles α and β, if the height of the tower is h, and the distance of the point of observation from the tower is d?

h cot α = d - cot β Signup and view all the answers

What is the formula for finding the distance d, in terms of angles α and β, and height h?

d = h(cot α - cot β) Signup and view all the answers

What is the formula for finding the length of the side x, in terms of angles α, β, and γ, and side a?

x = a(sin α sin β)/(sin γ cot β) Signup and view all the answers

What is the formula for finding the length of the side x, in terms of angles α, β, and γ, and side a, using sine rule?

x / sin β = a / sin α Signup and view all the answers

What is the angle of elevation of P as seen from O, if an observer is at O and the object is at P?

$\angle XOP$ Signup and view all the answers

What is the primary condition to be satisfied when drawing a figure for a heights and distances problem?

The figure should be drawn with all angles and distances labeled. Signup and view all the answers

What is the angle of depression of O as seen from P, if an observer is at P and the object is at O?

$\angle QPO$ Signup and view all the answers

What can be measured using trigonometry in heights and distances problems?

The heights of towers and temples, the distances between inaccessible points, and more. Signup and view all the answers

What is the relationship between a line and a plane in heights and distances problems?

A line is perpendicular to a plane if it is perpendicular to every line in that plane. Signup and view all the answers

What is the relationship between the median and the base in an isosceles triangle?

The median is perpendicular to the base Signup and view all the answers

What is the formula for finding the length of a side in a triangle, using cosine?

cos A = 2bc / sin A sin B sin C Signup and view all the answers

What is the condition for an observer to measure the bearing of an object?

The observer and object are on the same level Signup and view all the answers

What is the relationship between the interior and exterior angles of a triangle?

The exterior angle is equal to the sum of the interior opposite angles Signup and view all the answers

What is the name of the formula used to find the length of a side in a triangle, other than the right-angled triangle?

Cosine formula Signup and view all the answers

If the angle of elevation of a tower from a point is α and the angle of depression from the top of the tower is β, what is the correct expression for the height of the tower?

h cot α - cot β Signup and view all the answers

What is the correct expression for the length of the side x in terms of angles α, β, and γ, and side a, using the sine rule?

x = a sin β / sin γ Signup and view all the answers

If the angle of elevation of an object P from an observer O is α and the angle of depression from the object P is β, what is the correct expression for the distance d from the observer O to the point directly below the object P?

d = h cot α - cot β Signup and view all the answers

What is the correct expression for the height of a tower H, in terms of angles α and β, and height h?

H = h cot α - cot β Signup and view all the answers

What is the correct expression for the length of the side AB, in terms of angles α, β, and side a, using the sine rule?

AB = a sin β / sin α Signup and view all the answers

What is the condition for an observer at O to see an object at P at an angle of elevation?

The line of sight from O to P must be perpendicular to the horizontal line through O Signup and view all the answers

In a right-angled triangle, if one angle is 30 degrees, what is the measure of the other acute angle?

60 degrees Signup and view all the answers

What is the purpose of defining the angle of depression in heights and distances problems?

To measure the angle of an object seen from a lower level Signup and view all the answers

What is the characteristic of a right-angled triangle formed in heights and distances problems?

One side is perpendicular to the horizontal plane Signup and view all the answers

What is the significance of the horizontal line in defining the angle of elevation and depression?

It is the reference line for measuring angles Signup and view all the answers

Height and Distances

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Questions and Answers

In a triangle, what is the relationship between the internal bisector of an angle and the opposite side?

What is the name of the formula used to find the length of a side in a triangle, other than the right-angled triangle?

What is the relationship between the corresponding sides of similar triangles?

What is the bearing of an object P from an observer O, if the observer and the object are on the same level?

What is the exterior angle of a triangle equal to?

What is the formula for finding the height of a tower, where H = ?, in terms of angles α and β?

In a right-angled triangle, what is the relation between angles α and β, if the height of the tower is h, and the distance of the point of observation from the tower is d?

What is the formula for finding the distance d, in terms of angles α and β, and height h?

What is the formula for finding the length of the side x, in terms of angles α, β, and γ, and side a?

What is the formula for finding the length of the side x, in terms of angles α, β, and γ, and side a, using sine rule?

What is the angle of elevation of P as seen from O, if an observer is at O and the object is at P?

What is the primary condition to be satisfied when drawing a figure for a heights and distances problem?

What is the angle of depression of O as seen from P, if an observer is at P and the object is at O?

What can be measured using trigonometry in heights and distances problems?

What is the relationship between a line and a plane in heights and distances problems?

What is the relationship between the median and the base in an isosceles triangle?

What is the formula for finding the length of a side in a triangle, using cosine?

What is the condition for an observer to measure the bearing of an object?

What is the relationship between the interior and exterior angles of a triangle?

What is the name of the formula used to find the length of a side in a triangle, other than the right-angled triangle?

If the angle of elevation of a tower from a point is α and the angle of depression from the top of the tower is β, what is the correct expression for the height of the tower?

What is the correct expression for the length of the side x in terms of angles α, β, and γ, and side a, using the sine rule?

If the angle of elevation of an object P from an observer O is α and the angle of depression from the object P is β, what is the correct expression for the distance d from the observer O to the point directly below the object P?

What is the correct expression for the height of a tower H, in terms of angles α and β, and height h?

What is the correct expression for the length of the side AB, in terms of angles α, β, and side a, using the sine rule?

What is the condition for an observer at O to see an object at P at an angle of elevation?

In a right-angled triangle, if one angle is 30 degrees, what is the measure of the other acute angle?

What is the purpose of defining the angle of depression in heights and distances problems?

What is the characteristic of a right-angled triangle formed in heights and distances problems?

What is the significance of the horizontal line in defining the angle of elevation and depression?

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