G.5(D)·Geometry·Triangles & TrigonometryReadiness

Special Right Triangles and Elevation

Apply special right triangles and trigonometric ratios, including angles of elevation and depression.

Students model angles of elevation and depression with right triangles, solve special right triangles from known legs or hypotenuses, and recognize Pythagorean triples in distance and path diagrams.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Geometry released practice55 minutes

Learn the concept

Measure from the horizontal

An angle of elevation is measured upward from a horizontal line to a line of sight. An angle of depression is measured downward from a horizontal line. The line of sight connects the observer and the object.

Depression equals elevation

The horizontal line at the observer and the ground are parallel. The line of sight crosses both, so the angle of depression from the top equals the angle of elevation from the bottom as alternate interior angles.

Special right triangles

In a 45°-45°-90° triangle, the legs are equal and the hypotenuse is $s\sqrt{2}$ . In a 30°-60°-90° triangle, the shortest leg is s, the longer leg is $s\sqrt{3}$ , and the hypotenuse is $2s$ . Pythagorean triples such as 3-4-5 also give exact side lengths.

Find a leg from a 45°-45°-90° hypotenuse

The side pattern is $s,s,s\sqrt{2}$ . If the hypotenuse is h, each leg is $\frac{h}{\sqrt{2}}$ . For a hypotenuse of $12.4\sqrt{2}$ centimeters, each leg is $\frac{12.4\sqrt{2}}{\sqrt{2}}=12.4$ centimeters. Divide rather than multiply when working backward from the hypotenuse.

Follow a 30°-60°-90° path diagram

Match sides to their opposite angles: the short leg s faces 30°, the long leg $s\sqrt{3}$ faces 60°, and the hypotenuse $2s$ faces 90°. If the long leg is known, divide by $\sqrt{3}$ to find s. In a multi-triangle path diagram, solve one triangle first and transfer any shared length to the next triangle.

Pythagorean triples

Positive whole-number lengths a, b, c form a Pythagorean triple when $a^2+b^2=c^2$ , with c the hypotenuse. Common triples include 3-4-5, 5-12-13, and 8-15-17. Multiplying all three lengths by the same positive number preserves the right triangle; for example, 9-12-15 is three times 3-4-5.

Study released STAAR visuals

Use the original graphs, tables, and diagrams from released STAAR questions to practice reading labels, values, and relationships before answering the items below.

Archived released STAAR Geometry EOC · 2013 · Item 4 · 1 point

Point P stands atop a cliff and point Q is 32 feet from the cliff base. The angle between a horizontal ray from P and segment PQ is 45 degrees. The figure is not drawn to scale. — Angle of depression from P to QOriginal visual from the released STAAR test.

Archived released STAAR Geometry EOC · 2013 · Item 26 · 1 point

Three gears with centers A, B, and C. Segments AB and BC meet at a right angle at B, with A above B and C to the right of B. The distance from A to C is 9.1 times the square root of 2 centimeters. — Three gears forming a right triangleOriginal visual from the released STAAR test.

Archived released STAAR Geometry EOC · 2013 · Item 41 · 1 point

A square with right-angle marks at the top-right and bottom-left corners. A shaded path crosses it diagonally between two parallel 15-foot segments. The upper segment starts at the top-left corner, making a 30-degree angle with the top side, and meets the right side at a 60-degree angle. The lower segment starts on the left side, making a 60-degree angle with it, and ends at the bottom-right corner, making a 30-degree angle with the bottom side. The length y is marked on the right side from the end of the upper segment down to the bottom-right corner. — Shaded path across a squareOriginal visual from the released STAAR test.

Worked example

From the top of a 60-foot tower, the angle of depression to a car is 30°. How far is the car from the base of the tower?

The angle of elevation from the car to the tower top is also 30°.
The tower height is opposite the 30° angle, so it is the shortest leg: s = 60.
The ground distance is the longer leg: $60\sqrt{3}\approx 103.9$ feet.

$60\sqrt{3}\approx 103.9$ feet

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Geometry: Trigonometry

UT IPSI · Kid2Kid / Texas Education Agency

Use right-triangle relationships to solve for unknown distances in applied problems.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. A kite string makes a 45° angle with the ground, and the kite is 50 feet high. How long is the string?
Show answer and reasoning
$50\sqrt{2}\approx 70.7$ feet
The height and horizontal distance are equal legs, so the hypotenuse is $50\sqrt{2}$ .
2. The angle of depression from a cliff to a boat is 40°. What is the angle of elevation from the boat to the cliff top?
Show answer and reasoning
40°
The angles are alternate interior angles formed by parallel horizontal lines and the line of sight.
3. An isosceles right triangle has hypotenuse $12.4\sqrt{2}$ cm. Find each leg. A different right triangle has legs 9 cm and 12 cm; find its hypotenuse.
Show answer and reasoning
Each isosceles-triangle leg is 12.4 cm; the other hypotenuse is 15 cm.
Divide $12.4\sqrt{2}$ by $\sqrt{2}$ . For the other triangle, 9-12-15 is a multiple of 3-4-5, and $9^2+12^2=225=15^2$ .
4. A straight path is the hypotenuse of a 30°-60°-90° triangle. The path is 16 meters long. Find its shorter and longer perpendicular components.
Show answer and reasoning
8 m and $8\sqrt{3}$ m
Since $2s=16$ , the short leg is 8. The long leg is $s\sqrt{3}=8\sqrt{3}$ .

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 4 · 1 point

In the diagram, the angle of depression from $P$ to $Q$ is 45°. Which of the following is closest to the distance between $P$ and $Q$ ?

45.3 ft
22.6 ft
55.4 ft
18.5 ft

Show STAAR answer and explanation

45.3 ft

The 32-foot horizontal segment is adjacent to the 45° angle, and PQ is the hypotenuse. Thus $PQ=32/\cos45°\approx45.3$ feet.

22.6 ft: This is $32\sin45°$ , but PQ is the hypotenuse and must be longer than the 32-foot leg.

55.4 ft: This does not match $32/\cos45°$ ; the hypotenuse is approximately 45.3 feet.

18.5 ft: This is $32/\sqrt{3}$ , which does not apply to a 45° right triangle and is shorter than the given leg.

Ask the tutor about this question

Archived released STAAR Geometry EOC · 2013 · Item 26 · 1 point

Three gears in a machine are positioned relative to each other to form an isosceles right triangle, as shown. What is the distance in centimeters between the centers of the gears located at $B$ and $C$ ?

Show STAAR answer and explanation

9.1

A 45°–45°–90° triangle has hypotenuse equal to a leg times $\sqrt2$ . Thus $BC=(9.1\sqrt2)/\sqrt2=9.1$ cm.

18.2: Multiplying the hypotenuse by $\sqrt2$ instead of dividing gives 18.2.

Ask the tutor about this question

Archived released STAAR Geometry EOC · 2013 · Item 41 · 1 point

Within a square section of land, a landscaper will build a path, as represented by the shaded section in the diagram. Which measure is closest to the value of $y$ ?

5.5 ft
3.1 ft
4.3 ft
7.5 ft

Show STAAR answer and explanation

5.5 ft

Each unshaded triangle has hypotenuse 15. The square side is the long leg $15\cos30°=15\sqrt3/2$ . The short vertical leg is $15\sin30°=7.5$ . Therefore $y=15\sqrt3/2-7.5\approx5.49$ ft.

3.1 ft: This is not the difference of the long and short legs of the 15-ft special triangle.

4.3 ft: This does not use the full square side $15\sqrt3/2$ before subtracting 7.5.

7.5 ft: This is the short leg of an unshaded triangle, not the remaining vertical path segment.

Ask the tutor about this question

Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Draw and label angles of elevation and depression from a horizontal line.
Explain why an angle of depression equals the matching angle of elevation.
Apply 45°-45°-90° and 30°-60°-90° triangle patterns to find distances.
Recognize Pythagorean triples and their multiples to find exact missing lengths.

Instructional sequence

1
Engage
5 min
A person on a balcony looks down at a car. Ask students where the angle should be measured.
2
Direct instruction
12 min
Draw horizontal lines at both locations. Use alternate interior angles to show depression equals elevation.
3
Pattern review
15 min
Use $s,s,s\sqrt{2}$ and $s,s\sqrt{3},2s$ in both directions, then recognize Pythagorean triples and their multiples.
4
Guided practice
15 min
Sketch, label, and solve tower, drone, and kite problems, identifying the line-of-sight hypotenuse.
5
Closure
8 min
Students complete the exit ticket with a labeled sketch.

Misconceptions to address

““An angle of depression is measured from the vertical wall.””

Teacher move: Both elevation and depression are measured from a horizontal line of sight. Draw the horizontal before marking the angle.

““In a 45°-45°-90° triangle, the hypotenuse equals a leg.””

Teacher move: The hypotenuse is always the longest side. It is the leg length times $\sqrt{2}$ .

Materials

•Clinometer or protractor with string
•Scenario diagrams with horizontal lines
•Special right-triangle reference chart

Teaching tips

Require a sketch with both horizontal lines for every problem.
Label the line of sight as the hypotenuse before choosing a relationship.
Keep exact radical answers until the final rounding step.
Match each special-triangle side to its opposite angle; do not rely on the diagram's orientation.

Exit ticket · 3–5 min

A drone 30 meters above the ground sees a marker at a 45° angle of depression. Find the horizontal distance and the line-of-sight distance.

Show expected answer

The triangle is 45°-45°-90°, so the horizontal distance is 30 m. The line of sight is $30\sqrt{2}\approx 42.4$ m.

Sources & alignment

TEA 2013 Geometry released test TEA 2013 Geometry answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Measure from the horizontal

Depression equals elevation

Special right triangles

Find a leg from a 45°-45°-90° hypotenuse

Follow a 30°-60°-90° path diagram

Pythagorean triples

Study released STAAR visuals

Archived released STAAR Geometry EOC · 2013 · Item 4 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 26 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 41 · 1 point

Worked example

Watch the lesson

Geometry: Trigonometry

Try it yourself

1. A kite string makes a 45° angle with the ground, and the kite is 50 feet high. How long is the string?

2. The angle of depression from a cliff to a boat is 40°. What is the angle of elevation from the boat to the cliff top?

3. An isosceles right triangle has hypotenuse 12.4212.4\sqrt{2}12.42​ cm. Find each leg. A different right triangle has legs 9 cm and 12 cm; find its hypotenuse.

4. A straight path is the hypotenuse of a 30°-60°-90° triangle. The path is 16 meters long. Find its shorter and longer perpendicular components.

Released STAAR practice

Archived released STAAR Geometry EOC · 2013 · Item 4 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 26 · 1 point

Archived released STAAR Geometry EOC · 2013 · Item 41 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Pattern review

Guided practice

Closure

Misconceptions to address

3. An isosceles right triangle has hypotenuse $12.4\sqrt{2}$ cm. Find each leg. A different right triangle has legs 9 cm and 12 cm; find its hypotenuse.