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Prealgebra / LEVEL 13 · DIFFICULTY 4/5

Right Triangles & Quadrilaterals

Connect side lengths, angle structure and shape properties.

3 stages · 24 practice problems · two 6-question assessment forms

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  1. MINI QUEST 13.1The Pythagorean RelationshipRead the lesson
  2. MINI QUEST 13.2Special Right TrianglesRead the lesson
  3. MINI QUEST 13.3Quadrilateral Properties & DiagonalsRead the lesson
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STAGE 13.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

The Pythagorean Relationship

Useful preparation: Extracting Square Factors · Area & Composite Regions

Goal: Identify the hypotenuse and apply the right-triangle side equation.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

base baside c
The height meets the base at a right angle. In this right triangle, side c is the hypotenuse.

The hypotenuse is opposite the right angle

In a right triangle the hypotenuse is the longest side. Label it before using the side-length relationship.

a² + b² = c², c the hypotenuse

Solve for the missing squared length

To find a leg, subtract the other leg square from the hypotenuse square. Take a nonnegative root because a length is nonnegative.

a = √(c² − b²)

Check the condition in reverse

Three positive lengths form a right triangle if the square of the longest equals the sum of the other two squares. Diagonals in rectangles create right triangles.

rectangle diagonal = √(length² + width²)
WORKED EXAMPLE 1

A right triangle has legs 9 and 12. Find the hypotenuse.

  1. Square and add: 81 + 144 = 225.
  2. Take the positive square root.

15

WORKED EXAMPLE 2

A ladder 17 m long reaches a point 15 m above the ground. Its base and wall are perpendicular. How far is the base from the wall?

  1. The ladder is the hypotenuse.
  2. The horizontal distance is √(17² − 15²) = √64.

8 m

Common pitfalls

Possible mix-up: a + b = c for a right triangle.

The relation is between squares, not lengths.

Possible mix-up: Use the longest-looking side of a sketch as the hypotenuse.

Locate the stated right angle and its opposite side.

Explain it to yourself

Why does the hypotenuse have to be longer than either leg?

Preview the eight practice prompts
  1. A right triangle has legs 6 and 8. Find its hypotenuse.
  2. A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
  3. A rectangle is 7 by 24. Find the length of a diagonal.
  4. Do side lengths 8, 15 and 17 form a right triangle? Enter yes or no.
  5. Do side lengths 5, 6 and 8 form a right triangle? Enter yes or no.
  6. A right triangle has legs 4 and 4. Its hypotenuse is a√2. Find a.
  7. A 26 m cable joins the top of a vertical 24 m mast to a point on level ground. How far is that ground point from the mast base? New context
  8. A robot moves 12 units east and then 16 units north. How far is it directly from the start? New context
Open stage 13.1 in the student workspace →

STAGE 13.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Special Right Triangles

Useful preparation: The Pythagorean Relationship · Triangle & Polygon Angle Sums

Goal: Derive useful length ratios from symmetry and the Pythagorean relationship.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

A 45–45–90 triangle has equal legs

Symmetry gives two equal legs x. The hypotenuse square is 2x², so its length is x√2.

side ratio 1 : 1 : √2

Bisect an equilateral triangle

An altitude creates two 30–60–90 triangles. If the short leg is x, the hypotenuse is 2x and the other leg follows from the Pythagorean relationship.

side ratio 1 : √3 : 2

Match the side to its opposite angle

The side opposite 30° is shortest; the side opposite 90° is the hypotenuse. A ratio can be scaled only after identifying the correct side.

short leg = hypotenuse/2 in a 30–60–90 triangle
WORKED EXAMPLE 1

An isosceles right triangle has hypotenuse 10√2. Find each leg.

  1. The hypotenuse is a leg times √2.
  2. Divide by √2.

10

WORKED EXAMPLE 2

An equilateral triangle has side 12. Find its altitude.

  1. Half the base is 6, with hypotenuse 12.
  2. The altitude is 6√3 by the special-triangle ratio.

6√3

Common pitfalls

Possible mix-up: The longest leg is opposite 30°.

The shortest leg is opposite the smallest angle.

Possible mix-up: A 30–60–90 triangle has side ratio 1:2:3.

Its ratio is 1:√3:2.

Explain it to yourself

Derive the 30–60–90 ratio using one half of an equilateral triangle.

Preview the eight practice prompts
  1. An isosceles right triangle has legs 7. Its hypotenuse is a√2. Find a.
  2. A 45–45–90 triangle has hypotenuse 12√2. Find a leg.
  3. A 30–60–90 triangle has short leg 9. Find its hypotenuse.
  4. A 30–60–90 triangle has hypotenuse 28. Find its short leg.
  5. A 30–60–90 triangle has long leg 11√3. Find its short leg.
  6. An equilateral triangle has side 10. Its altitude is a√3. Find a.
  7. A square has diagonal 18√2 cm. What is its area in cm²? New context
  8. A triangular sign is equilateral with side 8 m. Its area is a√3 m². Find a. New context
Open stage 13.2 in the student workspace →

STAGE 13.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Quadrilateral Properties & Diagonals

Useful preparation: The Pythagorean Relationship · Parallel Lines & Transversals

Goal: Use definitions to distinguish universal properties from special cases.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

Classify by guaranteed properties

A parallelogram has both pairs of opposite sides parallel. A rectangle adds right angles, a rhombus adds four equal sides, and a square has both extra properties.

every square is both a rectangle and a rhombus

Diagonals behave differently by shape

Parallelogram diagonals bisect one another. Rectangle diagonals are also equal; rhombus diagonals are also perpendicular. None of these extra facts holds for every arbitrary quadrilateral.

parallelogram diagonals share a midpoint

Connect diagonals to area

Perpendicular diagonals partition a rhombus into four right triangles. Summing their areas gives half the product of the full diagonal lengths.

rhombus area = d₁d₂/2
WORKED EXAMPLE 1

A parallelogram diagonal has endpoints A and C and midpoint O. AO = 7.5. Find AC.

  1. The diagonal is bisected at O.
  2. Double AO.

15

WORKED EXAMPLE 2

A rhombus has diagonals 10 and 24. Find a side.

  1. Half-diagonals 5 and 12 form perpendicular legs.
  2. A side is √(5² + 12²).

13

Common pitfalls

Possible mix-up: Every rectangle has four equal sides.

That additional condition makes it a square.

Possible mix-up: All quadrilaterals have perpendicular diagonals.

Perpendicular diagonals require additional structure, such as a rhombus.

Explain it to yourself

Which properties make a square a member of more than one quadrilateral category?

Preview the eight practice prompts
  1. Is every square a rectangle? Enter yes or no.
  2. Is every rectangle a rhombus? Enter yes or no.
  3. A parallelogram has adjacent side lengths 8 and 13. Find its perimeter.
  4. A diagonal of a parallelogram is cut by the other diagonal into two pieces. One piece is 6.5. Find the full diagonal length.
  5. A rhombus has diagonals 14 and 20. Find its area.
  6. A parallelogram has one angle 68°. Find an adjacent angle.
  7. A rhombus-shaped kite has full diagonals 16 cm and 30 cm. Find the side length in cm. New context
  8. A rectangular display has diagonals of length 26. Their intersection is O. How far is O from any corner? New context
Open stage 13.3 in the student workspace →
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