# 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.

**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?

- Ayes
- Bno

What do you conclude about ?

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

- Ayes
- Bno

**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 ,

**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?

- ANo
- BYes

**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?

- ANo
- BYes

Hence, are the two lines perpendicular?

- AYes
- BNo

**Q16: **

In rectangle , suppose , , and . Is right angled?

- Ayes
- Bno

**Q17: **

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

- Ano
- Byes

**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 , not a right-angled triangle
- D , a right-angled triangle