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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

Choose an island to read its lesson.

MINI QUEST 13.1The Pythagorean RelationshipRead the lesson
MINI QUEST 13.2Special Right TrianglesRead the lesson
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.

NOTICE → MARK → REASON

Right triangle: find the useful sides

Mark the two legs, the right angle and the hypotenuse. This is a separate learning diagram; it does not supply the values for your practice question.

Highlight parts and add your information

Choose a part here, or select it in the diagram. Mark what you know, what you need, or a reason. These are your scratch notes; the website does not grade or save them.

One small move: Select a part above. Find its given measurement in the question, then label that part below.

Part of the diagramWhat does your mark mean?Length, angle or reasoning

Use the problem’s stated values. A drawing can look shorter after rotation even when the actual length stays the same. Changing the model or reloading clears these notes.

Go further: a diagonal inside a 3D box

TURN IT. LOOK INSIDE. EXPLAIN IT.

Rectangular prism · 3D explorer

Drag to rotate

A worked example you can change. Model values are separate from practice answers. Drag with a mouse or one finger. Use the turn buttons or arrow keys as an alternative.

Watch a quarter turn

$\displaystyle a = 4\, b = 3\, h = 6$

Dashed line: perpendicular height. Your marks use the colors you choose below.

Base length a: 4

Base width b: 3

Perpendicular height h: 6

Show a horizontal cross-sectionShow the two right triangles for the space diagonal

What should you notice?

Every horizontal layer has the same area. Doubling only the height doubles the number of layers and the volume.

Rotating the view changes what you see; it does not change the lengths or volume. Read the labelled dimensions, rather than measuring the screen.

Highlight parts and add your information

Choose a part here, or select it in the diagram. Mark what you know, what you need, or a reason. These are your scratch notes; the website does not grade or save them.

One small move: Select a part above. Find its given measurement in the question, then label that part below.

Part of the diagramWhat does your mark mean?Length, angle or reasoning

Use the problem’s stated values. A drawing can look shorter after rotation even when the actual length stays the same. Changing the model or reloading clears these notes.

From the model to the calculation

Find the base area $\displaystyle B = 4 \times 3 \approx 12$ square units.
V = Bh = $\displaystyle \left(4 \times 3\right) \times 6 \approx 72$ cubic units. Stacking equal-area layers explains the multiplication by height.
First find the base diagonal: $\displaystyle d^{2} = 4^{2} + 3^{2} = 25$ . It is perpendicular to the vertical edge. Then $\displaystyle D^{2} = d^{2} + 6^{2} = 61$ , so $\displaystyle D \approx 7.81$ . The space diagonal must be longer than any edge.
Check units: base area has square units; multiplying by a length gives cubic units. Rounded values are shown to three decimal places; keep $\displaystyle \pi$ or radicals for exact answers when requested.

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.

\displaystyle a^{2} + b^{2} = c^{2}

, 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.

\displaystyle a = \sqrt{c^{2} - b^{2}}

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.

Square and add: $\displaystyle 81 + 144 = 225$ .
Take the positive square root.

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?

The ladder is the hypotenuse.
The horizontal distance is $\displaystyle \sqrt{17^{2} - 15^{2}} = \sqrt{64}$ .

8 m

Common pitfalls

Possible mix-up: $\displaystyle a + b = c$ 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

Problem 1
A right triangle has legs 6 and 8. Find its hypotenuse.
Practise this exact problem →
Problem 2
A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
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Problem 3
A rectangle is 7 by 24. Find the length of a diagonal.
Practise this exact problem →
Problem 4
Do side lengths 8, 15 and 17 form a right triangle? Enter yes or no.
Practise this exact problem →
Problem 5
Do side lengths 5, 6 and 8 form a right triangle? Enter yes or no.
Practise this exact problem →
Problem 6
A right triangle has legs 4 and 4. Its hypotenuse is $\displaystyle a\, \sqrt{2}$ . Find a.
Practise this exact problem →
Problem 7New context
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?
Practise this exact problem →
Problem 8New context
A robot moves 12 units east and then 16 units north. How far is it directly from the start?
Practise this exact problem →

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 $\displaystyle 2\, x^{2}$ , so its length is $\displaystyle x\, \sqrt{2}$ .

side ratio

\displaystyle 1 : 1 : \sqrt{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

\displaystyle 1 : \sqrt{3} : 2

Match the side to its opposite angle

The side opposite $\displaystyle 30^{\circ}$ is shortest; the side opposite $\displaystyle 90^{\circ}$ 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 $\displaystyle 10\, \sqrt{2}$ . Find each leg.

The hypotenuse is a leg times $\displaystyle \sqrt{2}$ .
Divide by $\displaystyle \sqrt{2}$ .

WORKED EXAMPLE 2

An equilateral triangle has side 12. Find its altitude.

Half the base is 6, with hypotenuse 12.
The altitude is $\displaystyle 6\, \sqrt{3}$ by the special-triangle ratio.

$\displaystyle 6\, \sqrt{3}$

Common pitfalls

Possible mix-up: The longest leg is opposite $\displaystyle 30^{\circ}$ .

The shortest leg is opposite the smallest angle.

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

Its ratio is $\displaystyle 1 : \sqrt{3} : 2$ .

Explain it to yourself

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

Preview the eight practice prompts

Problem 1
An isosceles right triangle has legs 7. Its hypotenuse is $\displaystyle a\, \sqrt{2}$ . Find a.
Practise this exact problem →
Problem 2
A 45–45–90 triangle has hypotenuse $\displaystyle 12\, \sqrt{2}$ . Find a leg.
Practise this exact problem →
Problem 3
A 30–60–90 triangle has short leg 9. Find its hypotenuse.
Practise this exact problem →
Problem 4
A 30–60–90 triangle has hypotenuse 28. Find its short leg.
Practise this exact problem →
Problem 5
A 30–60–90 triangle has long leg $\displaystyle 11\, \sqrt{3}$ . Find its short leg.
Practise this exact problem →
Problem 6
An equilateral triangle has side 10. Its altitude is $\displaystyle a\, \sqrt{3}$ . Find a.
Practise this exact problem →
Problem 7New context
A square has diagonal $\displaystyle 18\, \sqrt{2}$ cm. What is its area in cm²?
Practise this exact problem →
Problem 8New context
A triangular sign is equilateral with side 8 m. Its area is $\displaystyle a\, \sqrt{3}\, m^{2}$ . Find a.
Practise this exact problem →

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 =

\displaystyle \frac{d_{1}\, d_{2}}{2}

WORKED EXAMPLE 1

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

The diagonal is bisected at O.
Double AO.

WORKED EXAMPLE 2

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

Half-diagonals 5 and 12 form perpendicular legs.
A side is $\displaystyle \sqrt{5^{2} + 12^{2}}$ .

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

Problem 1
Is every square a rectangle? Enter yes or no.
Practise this exact problem →
Problem 2
Is every rectangle a rhombus? Enter yes or no.
Practise this exact problem →
Problem 3
A parallelogram has adjacent side lengths 8 and 13. Find its perimeter.
Practise this exact problem →
Problem 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.
Practise this exact problem →
Problem 5
A rhombus has diagonals 14 and 20. Find its area.
Practise this exact problem →
Problem 6
A parallelogram has one angle $\displaystyle 68^{\circ}$ . Find an adjacent angle.
Practise this exact problem →
Problem 7New context
A rhombus-shaped kite has full diagonals 16 cm and 30 cm. Find the side length in cm.
Practise this exact problem →
Problem 8New context
A rectangular display has diagonals of length 26. Their intersection is O. How far is O from any corner?
Practise this exact problem →

Open stage 13.3 in the student workspace →

Browse another level →

Right Triangles & Quadrilaterals

The Pythagorean Relationship

Right triangle: find the useful sides

Rectangular prism · 3D explorer

From the model to the calculation

The hypotenuse is opposite the right angle

Solve for the missing squared length

Check the condition in reverse

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

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?

Common pitfalls

Explain it to yourself

Special Right Triangles

A 45–45–90 triangle has equal legs

Bisect an equilateral triangle

Match the side to its opposite angle

An isosceles right triangle has hypotenuse 10 2\displaystyle 10\, \sqrt{2}102​. Find each leg.

An equilateral triangle has side 12. Find its altitude.

Common pitfalls

Explain it to yourself

Quadrilateral Properties & Diagonals

Classify by guaranteed properties

Diagonals behave differently by shape

Connect diagonals to area

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

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

Common pitfalls

Explain it to yourself

An isosceles right triangle has hypotenuse $\displaystyle 10\, \sqrt{2}$ . Find each leg.

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