Worksheet: The Converse of the Pythagorean Theorem

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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?

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

Q2:

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

  • Ayes
  • Bno

Q3:

Is △ 𝐴 𝐶 𝐷 a right-angled triangle at 𝐶 ?

  • 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 𝑐 = 𝑎 + 𝑏 is necessarily a right triangle?

Let us assume that △ 𝐴 𝐵 𝐶 is of side lengths 𝑎 , 𝑏 , and 𝑐 , with 𝑐 = 𝑎 + 𝑏 . Let △ 𝐷 𝐵 𝐶 be a right triangle of side lengths 𝑎 , 𝑏 , and 𝑑 .

Using the Pythagorean theorem, what can you say about the relationship between 𝑎 , 𝑏 , and 𝑑 ?

  • A 𝑎 = 𝑑 + 𝑏
  • B 𝑏 = 𝑎 + 𝑑
  • C 𝑑 = 𝑎 + 𝑏

We know that for △ 𝐴 𝐵 𝐶 , 𝑐 = 𝑎 + 𝑏 .

What do you conclude about 𝑑 ?

  • A 𝑑 = 𝑐
  • B 𝑑 > 𝑐
  • C 𝑑 ≠ 𝑐

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

  • Ano
  • Byes

What do you conclude about △ 𝐴 𝐵 𝐶 ?

  • AIt is similar to △ 𝐷 𝐵 𝐶 , so it has a right angle at 𝐴 .
  • BIt is similar to △ 𝐷 𝐵 𝐶 , so it has a right angle at 𝐶 .
  • CIt is congruent to △ 𝐷 𝐵 𝐶 , so it has a right angle at 𝐶 .
  • DIt is congruent to △ 𝐷 𝐵 𝐶 , so it has a right angle at 𝐵 .
  • EIt is congruent to △ 𝐷 𝐵 𝐶 , so it has a right angle at 𝐴 .

Q6:

In triangle 𝐴 𝐵 𝐶 point 𝐷 lies on 𝐵 𝐶 and 𝐴 𝐷 ⟂ 𝐵 𝐶 , 𝐴 𝐶 = 3 7 . 8 , 𝐴 𝐷 = 1 0 . 0 8 , and 𝐴 𝐵 = 1 0 . 7 6 . Find the length of 𝐵 𝐶 to the nearest tenth, and then determine whether △ 𝐴 𝐵 𝐶 is a right-angled triangle or not.

  • A 𝐵 𝐶 = 3 5 . 4 , a right-angled triangle
  • B 𝐵 𝐶 = 2 . 9 , not a right-angled triangle
  • C 𝐵 𝐶 = 3 7 . 5 , a right-angled triangle
  • D 𝐵 𝐶 = 4 0 . 2 , not a right-angled triangle

Q7:

In triangle 𝐴 𝐵 𝐶 , let 𝐷 on 𝐵 𝐶 be the foot of the altitude from 𝐴 . If 𝐴 𝐶 = 1 1 8 . 9 , 𝐴 𝐷 = 6 9 . 6 1 8 , and 𝐵 𝐷 = 5 0 . 9 4 , is 𝐴 𝐵 𝐶 right angled at 𝐴 ?

  • Ano
  • Byes

Q8:

In triangle 𝐴 𝐵 𝐶 , 𝐴 𝐷 is perpendicular to 𝐵 𝐶 , 𝐷 lies between 𝐵 and 𝐶 , 𝐵 𝐷 = 8 , 𝐶 𝐷 = 2 , and 𝐴 𝐷 = 4 . Is 𝐴 𝐵 𝐶 a right-angled triangle?

  • Ano
  • Byes

Q9:

What does ( 𝐴 𝐶 ) equal to?

  • A ( 𝐶 𝐷 ) − ( 𝐴 𝐷 )
  • B ( 𝐶 𝐵 ) − ( 𝐴 𝐵 )
  • C 𝐶 𝐷 − 𝐷 𝐵
  • D 𝐶 𝐵 − 𝐴 𝐵

Q10:

In the figure shown, suppose that 𝐴 𝐸 = 2 𝐵 𝐶 and 𝐵 𝐷 = 8 . Determine 𝐴 𝐷 and 𝐸 𝐷 rounded to the nearest hundredth, if necessary.

  • A 𝐴 𝐷 = 8 . 8 c m , 𝐸 𝐷 = 1 7 . 7 4 c m
  • B 𝐴 𝐷 = 1 3 . 8 7 c m , 𝐸 𝐷 = 2 4 . 1 7 c m
  • C 𝐴 𝐷 = 8 . 8 c m , 𝐸 𝐷 = 2 1 . 6 7 c m
  • D 𝐴 𝐷 = 1 3 . 8 7 c m , 𝐸 𝐷 = 2 5 . 9 7 c m

Q11:

Is △ 𝐸 𝐴 𝐷 a right-angled triangle at 𝐴 ?

  • Ano
  • Byes

Q12:

𝐴 𝐵 𝐶 is a triangle where 𝐴 𝐵 = 3 c m , 𝐵 𝐶 = 4 c m and 𝐴 𝐶 = 5 c m . Find the size of ∠ 𝐴 𝐵 𝐶 .

Q13:

𝐴 𝐵 𝐶 is a triangle where 𝐴 𝐵 = 𝐴 𝐵 = 5 c m , 𝐵 𝐶 = 1 2 c m and 𝐴 𝐶 = 1 3 c m . Find the size of ∠ 𝐴 𝐵 𝐶 .

Q14:

Two lines intersect at the point 𝐴 ( 0 , 1 ) . One line goes through the point 𝐵 ( 2 , 3 ) , and the other goes through the point 𝐶 ( 2 , − 1 ) .

Find the lengths of 𝐴 𝐵 , 𝐴 𝐶 , and 𝐵 𝐶 .

  • A 𝐴 𝐵 = 2 √ 2 , 𝐴 𝐶 = 4 , 𝐵 𝐶 = 4
  • B 𝐴 𝐵 = 2 √ 2 , 𝐴 𝐶 = 2 √ 2 , 𝐵 𝐶 = 4
  • C 𝐴 𝐵 = 2 √ 2 , 𝐴 𝐶 = 2 √ 2 , 𝐵 𝐶 = 2 √ 2
  • D 𝐴 𝐵 = 4 , 𝐴 𝐶 = 2 √ 2 , 𝐵 𝐶 = 4
  • E 𝐴 𝐵 = 4 , 𝐴 𝐶 = 2 √ 2 , 𝐵 𝐶 = 2 √ 2

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

  • AYes
  • BNo

Are the two lines perpendicular?

  • AYes
  • BNo

Q15:

Two lines intersect at the point 𝐴 ( 3 , − 1 ) . One line goes through the point 𝐵 ( 5 , 1 ) , and the other goes through the point 𝐶 ( − 2 , 6 ) .

Find the lengths of 𝐴 𝐵 , 𝐴 𝐶 , and 𝐵 𝐶

  • A 𝐴 𝐵 = 2 √ 2 , 𝐴 𝐶 = √ 7 4 , 𝐵 𝐶 = √ 7 4
  • B 𝐴 𝐵 = 2 √ 2 , 𝐴 𝐶 = √ 7 4 , 𝐵 𝐶 = 2
  • C 𝐴 𝐵 = 2 √ 2 , 𝐴 𝐶 = √ 7 4 , 𝐵 𝐶 = 2 √ 2
  • D 𝐴 𝐵 = √ 7 4 , 𝐴 𝐶 = 2 √ 2 , 𝐵 𝐶 = √ 7 4
  • E 𝐴 𝐵 = 2 √ 2 , 𝐴 𝐶 = 2 √ 2 , 𝐵 𝐶 = √ 7 4

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

  • ANo
  • BYes

Hence, are the two lines perpendicular?

  • ANo
  • BYes

Q16:

In rectangle 𝐴 𝐵 𝐶 𝐷 , suppose 𝐴 𝐸 = 8 , 𝐷 𝐸 = 2 , and 𝐷 𝐶 = 4 . Is △ 𝐵 𝐸 𝐶 right angled?

  • Ano
  • Byes

Q17:

In the isosceles triangle 𝐴 𝐵 𝐶 below, 𝐴 𝐷 = 3 6 c m , 𝐵 𝐶 = 5 4 c m , and 𝐴 𝐸 = 6 0 c m . Is △ 𝐵 𝐴 𝐸 a right-angled triangle?

  • Ano
  • Byes

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?

  • Ano
  • Byes

Q20:

Is △ 𝐴 𝐶 𝐷 a right-angled triangle at 𝐶 ?

  • Ano
  • Byes

Q21:

Is △ 𝐴 𝐶 𝐷 a right-angled triangle at 𝐶 ?

  • Ano
  • Byes

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 𝐴 𝐵 𝐶 point 𝐷 lies on 𝐵 𝐶 and 𝐴 𝐷 ⟂ 𝐵 𝐶 , 𝐴 𝐶 = 1 0 . 5 , 𝐴 𝐷 = 6 , and 𝐴 𝐵 = 7 . 6 9 . Find the length of 𝐵 𝐶 to the nearest tenth, and then determine whether △ 𝐴 𝐵 𝐶 is a right-angled triangle or not.

  • A 𝐵 𝐶 = 1 2 . 5 , a right-angled triangle
  • B 𝐵 𝐶 = 4 . 4 , not a right-angled triangle
  • C 𝐵 𝐶 = 8 . 2 , a right-angled triangle
  • D 𝐵 𝐶 = 1 3 . 4 , not a right-angled triangle

Q25:

Is this triangle a right triangle?

  • AYes
  • BNo

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