G1 · Two Right Triangles in One Route

Alberta Grade 11 Mathematics

Geometry

Find missing lengths and angles by solving connected right triangles one step at a time.

What you will learn

• Identify right triangles and label their known sides and angles.
• Choose the Pythagorean theorem or a trigonometric ratio to find a missing measurement.
• Use a result from one triangle to solve another triangle.
• Include units, round sensibly, and check whether answers are reasonable.

Review: sides and right triangles

• The hypotenuse is opposite the right angle.
• Use the Pythagorean theorem when two side lengths are known.
• Use sine, cosine, or tangent when a known acute angle helps connect the sides.

Plan connected-triangle problems

• Mark each right angle and label the known measurements.
• Solve first the triangle that provides information needed later.
• Keep units consistent as you work.

Calculate, round, and check

• Use degree mode for angles stated in degrees.
• Round the final answer, not intermediate calculator values.
• Check the units and the physical meaning of the result.

Worked example

A route across a sloped work site

1. Find the first slanted length
   The horizontal distance and vertical rise are the legs of the first right triangle. The slanted side is its hypotenuse, so use the Pythagorean theorem.
   $$12^2+5^2=13^2$$
2. Use the result in the next triangle
   The first triangle's 13 m hypotenuse becomes a leg of the second triangle. Use it with the other leg, which is 12 m, to find the second hypotenuse.
   $$13^2+12^2=313$$
3. Report both lengths
   The first slanted length is 13 m. The second hypotenuse is the square root of 313 metres, which is about 17.7 m to the nearest tenth.
   $$\sqrt{313}\approx17.7$$

Worked example

Two ramps meeting at a platform

1. Record the first rise
   The first triangle already gives a vertical rise of 3 m. Its 4 m run confirms the triangle's horizontal leg, but is not needed to add the vertical rises.
   $$h_1=3\text{ m}$$
2. Find the second rise
   The second ramp is the hypotenuse. Its rise is opposite the $30^\circ$ angle, so sine relates the rise to the 6 m hypotenuse. The horizontal run is also positive: it is about 5.2 m.
   $$h_2=6\sin30^\circ=3\text{ m}$$
3. Add the vertical rises
   The second ramp starts at the top of the first. Therefore, the total rise is the sum of the two vertical rises, not the sum of the slanted ramp lengths.
   $$h_1+h_2=3+3=6\text{ m}$$

Worked example

A two-part cable support

1. Find the first cable length
   The mast and the ground distance are perpendicular legs. The cable is the hypotenuse, so use the Pythagorean theorem.
   $$\sqrt{8^2+6^2}=10\text{ m}$$
2. Find the second cable length
   The second triangle has a vertical leg of 8 m and a horizontal leg of 15 m. Its cable is the hypotenuse.
   $$\sqrt{8^2+15^2}=17\text{ m}$$
3. Find the second cable's angle
   For the angle above the horizontal, the vertical leg is opposite and the horizontal leg is adjacent. Inverse tangent gives the angle.
   $$\theta=\tan^{-1}\left(\frac{8}{15}\right)\approx28.1^\circ$$

Common mistakes and how to avoid them

Lesson summary

• A right triangle has one $90^\circ$ angle and a hypotenuse opposite it.
• Use the Pythagorean theorem to find a missing side when two side lengths are known.
• Use sine, cosine, or tangent to connect a known acute angle with side lengths.
• For connected triangles, solve the triangle that provides information needed next.
• Check units and whether each answer is reasonable.

Check your understanding

Key terms

Right triangle

A triangle with one angle measuring $90^\circ$.

Hypotenuse

The side opposite the right angle. It is the longest side of a right triangle.

Opposite side

For a chosen angle, the side directly across from that angle.

Adjacent side

For a chosen acute angle, the side beside it that is not the hypotenuse.

Trigonometric ratio

A comparison of two side lengths in a right triangle, such as sine, cosine, or tangent.

Continue through Math 20-3

View the complete Math 20-3 Alberta Grade 11 Mathematics curriculum and lessons

• M2 · Solve volume and capacity problems with SI and imperial units
• G2 · Solve practical scale and scale-diagram problems
• G3 · Model 3-D objects and draw their top, front, and side views
• G4 · Draw and describe simple exploded views and scaled components
• N1 · Use and verify strategies for numerical puzzles and games
• N3 · Explain and calculate compound-interest growth

Tutor support for G1 · Solve problems involving two or three right triangles

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برای مبحث G1 · مسئله‌های شامل دو یا سه مثلث قائم را حل کنید در درس Math 20-3، سؤال و تلاش خود را همراه بیاورید. در جلسه می‌توانید گام‌های تلاش خود را نشان دهید و آن را با یک نمونهٔ حل‌شده مقایسه کنید. راهنمای معلم ریاضی را ببینید یا جلسهٔ معارفهٔ رایگان ۳۰ دقیقه‌ای درخواست کنید. زمان جلسه پس از پذیرش معلم قطعی می‌شود.

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-3), expectation G1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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