# Worksheet: The Converse of the Pythagorean Theorem

In this worksheet, we will practice using the converse of the Pythagorean theorem to determine whether a triangle is a right triangle.

Q1:

What can the converse of the Pythagorean theorem be used for?

• Ademonstrating that a triangle is equilateral
• Bdemonstrating that a triangle has a right angle
• Cfinding the angles in a triangle
• Dfinding lengths in an equilateral triangle
• Edemonstrating that a triangle is an isosceles triangle

Q2:

Can the lengths 7.9 cm, 8.1 cm, and 5.3 cm form a right-angled triangle?

• Ano
• Byes

Q3:

Is a right-angled triangle at ? • Ayes
• Bno

Q4:

A triangle has sides of lengths 36.4, 27.3 and 45.5. What is its area?

Q5:

The Pythagorean theorem states that, in a right triangle, the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs. Does this mean that a triangle where is necessarily a right triangle?

Let us assume that is of side lengths , , and , with . Let be a right triangle of side lengths , , and . Using the Pythagorean theorem, what can you say about the relationship between , , and ?

• A
• B
• C

We know that for , .

What do you conclude about ?

• A
• B
• C

Is it possible to construct different triangles with the same length sides?

• Ano
• Byes

What do you conclude about ?

• AIt is similar to , so it has a right angle at .
• BIt is congruent to , so it has a right angle at .
• CIt is congruent to , so it has a right angle at .
• DIt is congruent to , so it has a right angle at .
• EIt is similar to , so it has a right angle at .

Q6:

In triangle point lies on and , , , and . Find the length of to the nearest tenth, and then determine whether is a right-angled triangle or not.

• A, a right-angled triangle
• B, not a right-angled triangle
• C, a right-angled triangle
• D, not a right-angled triangle

Q7:

In triangle , let on be the foot of the altitude from . If , , and , is right angled at ?

• Ano
• Byes

Q8:

In triangle , is perpendicular to , lies between and , , , and . Is a right-angled triangle? • Ano
• Byes

Q9:

What does equal to? • A
• B
• C
• D

Q10:

In the figure shown, suppose that and . Determine and rounded to the nearest hundredth, if necessary. • A,
• B,
• C,
• D,

Q11:

Is a right-angled triangle at ? • Ano
• Byes

Q12:

is a triangle where , and . Find the size of .

Q13:

is a triangle where , and . Find the size of .

Q14:

Two lines intersect at the point . One line goes through the point , and the other goes through the point .

Find the lengths of , , and .

• A, ,
• B, ,
• C, ,
• D, ,
• E, ,

Using the Pythagorean theorem, decide: is triangle a right triangle?

• AYes
• BNo

Are the two lines perpendicular?

• AYes
• BNo

Q15:

Two lines intersect at the point . One line goes through the point , and the other goes through the point .

Find the lengths of , , and

• A, ,
• B, ,
• C, ,
• D, ,
• E, ,

Using the Pythagorean theorem, decide: is triangle a right triangle?

• AYes
• BNo

Hence, are the two lines perpendicular?

• AYes
• BNo

Q16:

In rectangle , suppose , , and . Is right angled? • Ano
• Byes

Q17:

In the isosceles triangle below, , , and . Is a right-angled triangle? • Ayes
• Bno

Q18:

Can the lengths 16.6 cm, 6.3 cm, and 11.3 cm form a right-angled triangle?

• Ano
• Byes

Q19:

Can the lengths 14.4 cm, 19.2 cm, and 24 cm form a right-angled triangle?

• Ayes
• Bno

Q20:

Is a right-angled triangle at ? • Ano
• Byes

Q21:

Is a right-angled triangle at ? • Ayes
• Bno

Q22:

A triangle has sides of lengths 20.4, 59.5 and 62.9. What is its area?

Q23:

A triangle has sides of lengths 44, 4.2 and 44.2. What is its area?

Q24:

In triangle point lies on and , , , and . Find the length of to the nearest tenth, and then determine whether is a right-angled triangle or not.

• A, a right-angled triangle
• B, not a right-angled triangle
• C, a right-angled triangle
• D, not a right-angled triangle

Q25:

Is this triangle a right triangle? • AYes
• BNo