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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:

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 = π‘Ž + 𝑏 2 2 2 is necessarily a right triangle?

Let us assume that β–³ 𝐴 𝐡 𝐢 is of side lengths π‘Ž , 𝑏 , and 𝑐 , with 𝑐 = π‘Ž + 𝑏 2 2 2 . Let β–³ 𝐷 𝐡 𝐢 be a right triangle of side lengths π‘Ž , 𝑏 , and 𝑑 .

Using the Pythagorean theorem, what can you say about the relationship between π‘Ž , 𝑏 , and 𝑑 ?

  • A 𝑏 = π‘Ž + 𝑑 2 2 2
  • B π‘Ž = 𝑑 + 𝑏 2 2 2
  • C 𝑑 = π‘Ž + 𝑏 2 2 2

We know that for β–³ 𝐴 𝐡 𝐢 , 𝑐 = π‘Ž + 𝑏 2 2 2 .

What do you conclude about 𝑑 ?

  • A 𝑑 = 𝑐
  • B 𝑑 > 𝑐
  • C 𝑑 β‰  𝑐

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

  • Ayes
  • Bno

What do you conclude about β–³ 𝐴 𝐡 𝐢 ?

  • AIt is congruent to β–³ 𝐷 𝐡 𝐢 , so it has a right angle at 𝐢 .
  • BIt is congruent to β–³ 𝐷 𝐡 𝐢 , so it has a right angle at 𝐡 .
  • CIt is similar to β–³ 𝐷 𝐡 𝐢 , so it has a right angle at 𝐢 .
  • DIt is similar to β–³ 𝐷 𝐡 𝐢 , so it has a right angle at 𝐴 .
  • EIt is congruent to β–³ 𝐷 𝐡 𝐢 , so it has a right angle at 𝐴 .

Q2:

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

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

Q3:

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

  • Ano
  • Byes

Q4:

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

  • Ano
  • Byes

Q5:

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

  • Ayes
  • Bno

Q6:

Is a right triangle at ?

  • Ano
  • Byes

Q7:

Is a right triangle at ?

  • Ayes
  • Bno

Q8:

Is a right triangle at ?

  • Ano
  • Byes

Q9:

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

Q10:

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

Q11:

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

Q12:

In triangle , is perpendicular to , lies between and , , , and . Is a right triangle?

  • Ayes
  • Bno

Q13:

What does ( 𝐴 𝐢 ) 2 equal to?

  • A 𝐢 𝐡 βˆ’ 𝐴 𝐡
  • B ( 𝐢 𝐷 ) βˆ’ ( 𝐴 𝐷 ) 2 2
  • C 𝐢 𝐷 βˆ’ 𝐷 𝐡
  • D ( 𝐢 𝐡 ) βˆ’ ( 𝐴 𝐡 ) 2 2

Q14:

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

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

Q15:

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

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

Q16:

Two lines intersect at the point 𝐴 ( 0 , 1 ) . One line goes through the point 𝐡 ( 2 , 3 ) , and the other goes through the point 𝐢 ( 2 , βˆ’ 1 ) .

Find the lengths of 𝐴 𝐡 , 𝐴 𝐢 , and 𝐡 𝐢 .

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

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

  • AYes
  • BNo

Are the two lines perpendicular?

  • ANo
  • BYes

Q17:

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

  • Ano
  • Byes

Q18:

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

  • Ayes
  • Bno

Q19:

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

  • Ano
  • Byes

Q20:

Two lines intersect at the point 𝐴 ( 3 , βˆ’ 1 ) . One line goes through the point 𝐡 ( 5 , 1 ) , and the other goes through the point 𝐢 ( βˆ’ 2 , 6 ) .

Find the lengths of 𝐴 𝐡 , 𝐴 𝐢 , and 𝐡 𝐢 .

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

According to the Pythagorean theorem, is triangle 𝐴 𝐡 𝐢 a right triangle?

  • ANo
  • BYes

Hence, are the two lines perpendicular?

  • AYes
  • BNo

Q21:

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

Q22:

In the figure shown, suppose that 𝐴 𝐸 = 2 𝐡 𝐢 and 𝐡 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 = 1 3 . 8 7 c m , 𝐸 𝐷 = 2 5 . 9 7 c m
  • D 𝐴 𝐷 = 8 . 8 c m , 𝐸 𝐷 = 2 1 . 6 7 c m

Q23:

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

Q24:

Is a right triangle at ?

  • Ano
  • Byes

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