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Heights and Distances Final Assessment

Core Concepts

Question 1mcq
In a second review item, from two points on the same straight line with a tower, angles of elevation are 30∘30^\circ and 60∘60^\circ. What is the key modelling idea?
A.Use only the larger angle as the answer
B.Use two right triangles sharing the same vertical height
C.Use two separate heights
D.Ignore the nearer observation
Question 2mcq
In a second review item, two observation points are on the same side of a tower, and the nearer point is 20 m from the farther point. Which base lengths are related correctly?
A.Farther distance = nearer distance + 20
B.Nearer distance = farther distance + 20
C.Both distances are equal
D.The 20 m is the tower height
Question 3mcq
In a second review item, two observers stand on opposite sides of a tower. What does the distance between observers equal?
A.Difference of their distances
B.Height of the tower
C.Hypotenuse of one triangle
D.Sum of their distances from the foot of the tower
Question 4mcq
In a second review item, which standard value is useful for a height problem with angle 60∘60^\circ and base 1010 m?
A.sin⁡60∘=12\sin60^\circ=\frac12
B.cos⁡60∘=3\cos60^\circ=\sqrt3
C.tan⁡60∘=3\tan60^\circ=\sqrt3
D.tan⁡60∘=1\tan60^\circ=1
Question 5mcq
In a second review item, a student gets a tower height of 22 m when the tower is viewed from 10 m away at 60∘60^\circ. What should they notice?
A.Units are not needed
B.The answer is unreasonable because a 60∘60^\circ angle from 10 m suggests height greater than 10 m
C.The answer must be correct because it is positive
D.All height answers must be less than distance
Question 6mcq
In a second review item, a student treats the angle of depression from a cliff to a boat as the angle at the boat below the horizontal. What is the error?
A.The angle of depression is measured from the observer’s horizontal at the top
B.Depression angles are measured from the vertical
C.The boat angle is always 90∘90^\circ
D.Angle of depression cannot equal angle of elevation

Skill Practice

Question 7mcq
In a second review item, which choice best checks the skill: identify angle of elevation?
A.Ignore the horizontal line of sight
B.Use the hypotenuse as height every time
C.Choose a formula before identifying known sides
D.Draw or choose a right-triangle model, then use the correct angle, side and unit information.
Question 8mcq
In a second review item, which choice best checks the skill: identify angle of depression?
A.Use the hypotenuse as height every time
B.Choose a formula before identifying known sides
C.Draw or choose a right-triangle model, then use the correct angle, side and unit information.
D.Ignore the horizontal line of sight
Question 9mcq
In a second review item, which choice best checks the skill: draw diagrams from word problems?
A.Choose a formula before identifying known sides
B.Draw or choose a right-triangle model, then use the correct angle, side and unit information.
C.Ignore the horizontal line of sight
D.Use the hypotenuse as height every time
Question 10mcq
In a second review item, which choice best checks the skill: choose the correct trigonometric ratio?
A.Draw or choose a right-triangle model, then use the correct angle, side and unit information.
B.Ignore the horizontal line of sight
C.Use the hypotenuse as height every time
D.Choose a formula before identifying known sides
Question 11mcq
From a point 12 m away, the angle of elevation of a pole is 30∘30^\circ. Find the height.
A.1212 m
B.12212\sqrt2 m
C.2424 m
D.434\sqrt3 m
Question 12mcq
A tower is 3636 m high and the angle of elevation is 45∘45^\circ. Find the horizontal distance.
A.7272 m
B.36336\sqrt3 m
C.3636 m
D.3636 m

Application and Reasoning

Question 13mcq
In a second review item, which choice best checks the skill: solve problems with two observations?
A.Choose a formula before identifying known sides
B.Draw or choose a right-triangle model, then use the correct angle, side and unit information.
C.Ignore the horizontal line of sight
D.Use the hypotenuse as height every time
Question 14mcq
In a second review item, which choice best checks the skill: handle same-side observations?
A.Draw or choose a right-triangle model, then use the correct angle, side and unit information.
B.Ignore the horizontal line of sight
C.Use the hypotenuse as height every time
D.Choose a formula before identifying known sides
Question 15mcq
In a second review item, which choice best checks the skill: handle opposite-side observations?
A.Ignore the horizontal line of sight
B.Use the hypotenuse as height every time
C.Choose a formula before identifying known sides
D.Draw or choose a right-triangle model, then use the correct angle, side and unit information.
Question 16mcq
In a second review item, which choice best checks the skill: use 30°, 45° and 60° values?
A.Use the hypotenuse as height every time
B.Choose a formula before identifying known sides
C.Draw or choose a right-triangle model, then use the correct angle, side and unit information.
D.Ignore the horizontal line of sight
Question 17mcq
In a second review item, which choice best checks the skill: check reasonableness of height-distance answers?
A.Choose a formula before identifying known sides
B.Draw or choose a right-triangle model, then use the correct angle, side and unit information.
C.Ignore the horizontal line of sight
D.Use the hypotenuse as height every time
Question 18mcq
In a second review item, which choice best checks the skill: avoid elevation-depression diagram errors?
A.Draw or choose a right-triangle model, then use the correct angle, side and unit information.
B.Ignore the horizontal line of sight
C.Use the hypotenuse as height every time
D.Choose a formula before identifying known sides

Mixed Challenge

Question 19mcq
In a second review item, a student looking up from the ground to the top of a tower forms which angle with the horizontal line through the eye?
A.Angle of depression
B.Right angle only
C.Angle at the top of the tower
D.Angle of elevation
Question 20mcq
In a second review item, from the top of a lighthouse, an observer looks down at a boat. The angle made with the horizontal through the observer is called:
A.Central angle
B.Angle of rotation
C.Angle of depression
D.Angle of elevation
Question 21mcq
In a second review item, a 20 m tree casts a shadow, and the angle of elevation of the sun is 45∘45^\circ. Which diagram model is most appropriate?
A.A triangle with the shadow as hypotenuse
B.A right triangle with tree as vertical height and shadow as horizontal base
C.A circle with the tree as radius
D.Two unrelated right triangles
Question 22mcq
In a second review item, a tower height is unknown, the horizontal distance from its foot is known, and the angle of elevation is given. Which ratio is usually most direct?
A.tan⁡θ=heightdistance\tan\theta=\frac{\text{height}}{\text{distance}}
B.sin⁡θ=distanceheight\sin\theta=\frac{\text{distance}}{\text{height}}
C.cos⁡θ=heightdistance\cos\theta=\frac{\text{height}}{\text{distance}}
D.sec⁡θ=heighthypotenuse\sec\theta=\frac{\text{height}}{\text{hypotenuse}}
Question 23mcq
In a second review item, a man stands 30 m from a tower. The angle of elevation of the top is 30∘30^\circ. What is the tower height?
A.30330\sqrt{3} m
B.1515 m
C.6060 m
D.10310\sqrt{3} m
Question 24mcq
In a second review item, a building is 24 m high. The angle of elevation of its top from a point on the ground is 45∘45^\circ. What is the distance from the building?
A.24224\sqrt2 m
B.4848 m
C.2424 m
D.1212 m