Solving Right Triangles
Ximena wants to know how tall the pyramid she visited at Teotihuacan is, but she can't climb up with a tape measure. She can, however, measure a distance on the ground and the angle at which she sees the top. Can you find a height without touching it? With a right triangle, you can.
Solving a right triangle
"Solving" a triangle means finding all of its sides and angles. In a right triangle you already know the right angle: you only need two more pieces of data (two sides, or one side and an acute angle).
Remember the ratios for an acute angle $\alpha$: $\sin\alpha=\dfrac{\text{opposite leg}}{\text{hypotenuse}}$, $\cos\alpha=\dfrac{\text{adjacent leg}}{\text{hypotenuse}}$, $\tan\alpha=\dfrac{\text{opposite leg}}{\text{adjacent leg}}$.
Constructing the triangle
To draw a right triangle with a ruler, a set square, and a protractor: draw one leg, raise a perpendicular at one of its endpoints, and, if you have an acute angle, measure it from the other endpoint until the side meets the perpendicular. If you have both legs, just join the endpoints. A sketch with the data labeled is always the first step in solving.
A right-angled triangle has legs measuring $5$ cm and $12$ cm. Calculate the length of the hypotenuse and the measure in degrees of its two acute angles.
A right-angled triangle has legs measuring $5$ cm and $12$ cm. Calculate the length of the hypotenuse and the measure in degrees of its two acute angles.
Points $A(0,0)$, $B(8,0)$, and $C(8,6)$ form a right-angled triangle in the Cartesian plane. Calculate the trigonometric ratios (sine, cosine, and tangent) of the angle…
Points $A(0,0)$, $B(8,0)$, and $C(8,6)$ form a right-angled triangle in the Cartesian plane. Calculate the trigonometric ratios (sine, cosine, and tangent) of the angle at vertex $A$.
A ladder $4$ m in length is resting against a vertical wall. If the foot of the ladder is $1.2$ m from the wall, determine the angle the ladder makes with the horizontal…
A ladder $4$ m in length is resting against a vertical wall. If the foot of the ladder is $1.2$ m from the wall, determine the angle the ladder makes with the horizontal ground and the height reached by its top end on the wall.
Sides with unknowns: Pythagoras and a quadratic equation
Sometimes the problem doesn't give the sides as numbers, but in terms of an unknown $x$. Then Pythagoras becomes an equation you have to solve. After that you get the exact trigonometric ratios as fractions, without a calculator.
In a right-angled triangle, the length of one leg is $x$ cm, the other leg is $(x+7)$ cm and the hypotenuse is $(x+14)$ cm long. Set up and solve the corresponding…
In a right-angled triangle, the length of one leg is $x$ cm, the other leg is $(x+7)$ cm and the hypotenuse is $(x+14)$ cm long. Set up and solve the corresponding equation to determine the value of $x$. Then, calculate the exact value (as an irreducible fraction) of the sine and cosine of the smallest angle in the triangle.
Inaccessible heights: angle of elevation
The angle of elevation is the angle the line of sight to a high point makes with the horizontal. It is the classic way to measure a height without climbing it.
An obelisk casts a shadow $15$ m long on the horizontal ground at the moment when the sun's rays form an angle of $42^\circ$ with the horizontal. Calculate the height of…
An obelisk casts a shadow $15$ m long on the horizontal ground at the moment when the sun's rays form an angle of $42^\circ$ with the horizontal. Calculate the height of the obelisk.
A child flies a kite attached to a string of length $45$ m, which is held taut. If the string forms an angle of $55^\circ$ with the horizontal ground, at what height is…
A child flies a kite attached to a string of length $45$ m, which is held taut. If the string forms an angle of $55^\circ$ with the horizontal ground, at what height is the kite? (Assume the child holds the string at ground level).
On a rectangular computer screen, the diagonal measures $60$ cm and forms an angle of $29^\circ$ with the bottom edge of the frame. Calculate the width and height of the…
On a rectangular computer screen, the diagonal measures $60$ cm and forms an angle of $29^\circ$ with the bottom edge of the frame. Calculate the width and height of the screen.
A straight wheelchair ramp has a length of $6$ m and covers a vertical rise of $0.45$ m. Determine the angle of inclination of the ramp relative to the horizontal.
A straight wheelchair ramp has a length of $6$ m and covers a vertical rise of $0.45$ m. Determine the angle of inclination of the ramp relative to the horizontal.
Angle of depression
If you look down from a high point, the angle between the horizontal and your line of sight is the angle of depression. By alternate interior angles between parallel lines, it is equal to the angle of elevation seen from the low point.
From the lantern room of a lighthouse, located $45$ m above sea level, two ships are observed sailing in line with the base of the lighthouse. The angle of depression to…
From the lantern room of a lighthouse, located $45$ m above sea level, two ships are observed sailing in line with the base of the lighthouse. The angle of depression to the nearer ship is $34^\circ$, and to the farther ship is $18^\circ$. Calculate the exact distance separating the two ships.
Two vertical industrial chimneys of heights $20$ m and $35$ m are built on flat ground. From the midpoint of the line segment joining their bases, the top of the first…
Two vertical industrial chimneys of heights $20$ m and $35$ m are built on flat ground. From the midpoint of the line segment joining their bases, the top of the first chimney is observed with an angle of elevation of $35^\circ$. What will be the angle of elevation of the top of the second chimney from the same midpoint?
Two observation points: a system of equations
If you can't measure the distance to the base, observe from two points in line with it and set up a system with the same height $h$.
This is a system of two equations with two unknowns ($h$ and $x$), which you can solve by substitution, elimination by equating, or elimination by addition.
From a point on the ground, the nacelle of a wind turbine is observed with an angle of elevation of $32^\circ$. Moving $25$ m in a straight line towards the base of the…
From a point on the ground, the nacelle of a wind turbine is observed with an angle of elevation of $32^\circ$. Moving $25$ m in a straight line towards the base of the tower, the new angle of elevation is $50^\circ$. Set up a system of trigonometric equations and calculate the height of the wind turbine.
Two observers, $A$ and $B$, are separated by a distance of $300$ m on a horizontal plain. Both look towards a stationary drone that is located exactly in the vertical…
Two observers, $A$ and $B$, are separated by a distance of $300$ m on a horizontal plain. Both look towards a stationary drone that is located exactly in the vertical plane passing through the straight line joining them, positioned between them. Observer $A$ sees the drone with an angle of elevation of $60^\circ$ and observer $B$ sees it with an angle of $45^\circ$. Calculate the exact altitude at which the drone is flying, expressing the result in surd form.
A vertical post casts a shadow on flat ground. At a point in the morning, when the sun's angle of elevation is $\alpha = 30^\circ$, the shadow measures $L$ meters. A…
A vertical post casts a shadow on flat ground. At a point in the morning, when the sun's angle of elevation is $\alpha = 30^\circ$, the shadow measures $L$ meters. A couple of hours later, when the sun's angle of elevation has increased to $\beta = 45^\circ$, it is found that the shadow has shortened by exactly $10$ meters. Set up a system of equations and calculate the exact height of the post expressed in radical form.
Bearings and crossing paths
In navigation and aviation, direction is given with an angle measured from North (or from the North-South line). For example, "N $40^\circ$ E" means: start at North and turn $40^\circ$ toward East. North is up, East is to the right.
If two vehicles leave the same point at constant speed, the distance traveled is $d=v\,t$. If their bearings form a right angle (one heading North and the other East), their positions are the two legs of a right triangle. To find how one sees the other, measure the angle at the vertex where the observer is, relative to the North line passing through it.
A coastguard detects that a ship $A$ is sailing in a straight line towards the East at a constant speed of $20$ knots, and a ship $B$ is sailing towards the North at…
A coastguard detects that a ship $A$ is sailing in a straight line towards the East at a constant speed of $20$ knots, and a ship $B$ is sailing towards the North at $15$ knots. At a given moment ($t=0$), both ships are at exactly the same point, where their paths cross. At what angle relative to North will the captain of ship $A$ observe ship $B$ exactly $2$ hours after crossing?
Figures that split into right triangles
Ramps, rhombuses, regular polygons, tangents to a circle, and isosceles triangles are solved by drawing an altitude or a radius that divides them into right triangles.
An isosceles triangle has two equal sides measuring $14$ cm each, and the non-equal angle (the included angle between them) measures $40^\circ$. By drawing the altitude…
An isosceles triangle has two equal sides measuring $14$ cm each, and the non-equal angle (the included angle between them) measures $40^\circ$. By drawing the altitude to the base to divide it into two right-angled triangles, calculate the length of that base.
Calculate the area and perimetre of a regular hexagon inscribed in a circle of radius $12$ cm. (Remember to divide the polygon into triangles to use trigonometry).
Calculate the area and perimetre of a regular hexagon inscribed in a circle of radius $12$ cm. (Remember to divide the polygon into triangles to use trigonometry).
The diagonals of a rhombus measure $24$ cm and $10$ cm. Given that the diagonals of a rhombus bisect each other at right angles, calculate the length of the sides of the…
The diagonals of a rhombus measure $24$ cm and $10$ cm. Given that the diagonals of a rhombus bisect each other at right angles, calculate the length of the sides of the rhombus and the size of its four interior angles.
Common chord of two circles
When two circles intersect at two points, the segment joining those points is their common chord. The line joining the centers is perpendicular to that chord and meets it at its midpoint $M$. That is why two right triangles appear: in each one, the hypotenuse is a radius and one leg is half the chord.
The tangent to a circle is perpendicular to its radius. So the angle between the tangents at $P$ is equal to the angle between the radii $PO_1$ and $PO_2$ (or its supplement, depending on which of the two angles is asked for).
Two intersecting circles have radii of $10$ cm and $17$ cm. The length of the common chord of both circles is $16$ cm, and the centers lie on opposite sides of this…
Two intersecting circles have radii of $10$ cm and $17$ cm. The length of the common chord of both circles is $16$ cm, and the centers lie on opposite sides of this chord. Calculate the distance between the centers of the two circles and determine the angle formed between their respective tangent lines at one of the points of intersection.
Slopes, tangents, and challenge problems
The slope of a road is given as a percentage: $15\%$ means rising $15$ m for every $100$ m horizontally, that is, $\tan\alpha=0.15$. A tangent to a circle is perpendicular to the radius at the point of contact, which produces a right triangle.
When the problem asks you to "prove," use only the definitions: with a triangle of angle $\alpha$, height $h$, base $b=\dfrac{h}{\tan\alpha}$, and ramp $r=\dfrac{h}{\sin\alpha}$, write $b+r$ with the common denominator $\sin\alpha$ and you'll see $\cos\alpha$ appear.
A road sign warns of a steep gradient of $15\%$. This indicates that for every $100$ m travelled in the horizontal projection, there is a vertical rise of $15$ m.…
A road sign warns of a steep gradient of $15\%$. This indicates that for every $100$ m travelled in the horizontal projection, there is a vertical rise of $15$ m. Calculate the angle of inclination of the road and the actual distance a vehicle will travel on the road surface if it advances $600$ m in the horizontal projection.
From a point $P$ outside a circle of radius $8$ cm, two tangent lines to the circle are drawn. If the distance from point $P$ to the center of the circle is $17$ cm,…
From a point $P$ outside a circle of radius $8$ cm, two tangent lines to the circle are drawn. If the distance from point $P$ to the center of the circle is $17$ cm, calculate the total angle between the two tangent lines.
A rectangular solar panel $2.5$ m long is inclined at $35^\circ$ to the horizontal on a flat roof. Calculate the length of the shadow cast by the panel on the roof when…
A rectangular solar panel $2.5$ m long is inclined at $35^\circ$ to the horizontal on a flat roof. Calculate the length of the shadow cast by the panel on the roof when the sun's rays strike completely perpendicular to the plane of the roof.
A telecommunications antenna is installed on the flat roof of a building. From a point on the ground, situated $45$ m from the base of the building, a surveyor observes…
A telecommunications antenna is installed on the flat roof of a building. From a point on the ground, situated $45$ m from the base of the building, a surveyor observes the base of the antenna with an angle of elevation of $38^\circ$ and the top of the antenna with an angle of elevation of $46^\circ$. Calculate the height of the building and the exact length of the antenna.
It is desired to construct a straight ramp to overcome a vertical drop of height $h$. If the angle of inclination of the ramp relative to the horizontal is $\alpha$,…
It is desired to construct a straight ramp to overcome a vertical drop of height $h$. If the angle of inclination of the ramp relative to the horizontal is $\alpha$, prove using the definitions of trigonometric ratios that the sum of the length of the ramp's base (its horizontal projection) plus the length of the ramp itself is equal to $h \times \frac{1 + \cos(\alpha)}{\sin(\alpha)}$.
In an isosceles triangle whose base is $12$ cm long and whose altitude above this base is $18$ cm, a rectangle is inscribed such that one of its sides lies on the base…
In an isosceles triangle whose base is $12$ cm long and whose altitude above this base is $18$ cm, a rectangle is inscribed such that one of its sides lies on the base of the triangle and the other two vertices touch the equal sides. If the diagonals of this rectangle intersect forming an angle of $60^\circ$ (this being the angle subtending the base of the rectangle), set up the trigonometric and geometric system required to calculate the exact dimensions of this rectangle.
Back to the beginning: Ximena measures a distance on the ground to the base of the pyramid, uses a clinometer to measure the angle of elevation of the top, and calculates $h=d\tan\alpha$ (adding the height of her eyes).