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?

  • 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 △𝐴𝐶𝐷 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 △𝐴𝐵𝐶 is of side lengths 𝑎, 𝑏, and 𝑐, with 𝑐=𝑎+𝑏. Let △𝐷𝐵𝐶 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?

  • A𝑎=𝑑+𝑏
  • B𝑏=𝑎+𝑑
  • C𝑑=𝑎+𝑏

We know that for △𝐴𝐵𝐶, 𝑐=𝑎+𝑏.

What do you conclude about 𝑑?

  • A𝑑≠𝑐
  • B𝑑=𝑐
  • C𝑑>𝑐

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

  • Ano
  • Byes

What do you conclude about △𝐴𝐵𝐶?

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

Q6:

In triangle 𝐴𝐵𝐶 point 𝐷 lies on 𝐵𝐶 and 𝐴𝐷⟂𝐵𝐶, 𝐴𝐶=37.8, 𝐴𝐷=10.08, and 𝐴𝐵=10.76. Find the length of 𝐵𝐶 to the nearest tenth, and then determine whether △𝐴𝐵𝐶 is a right-angled triangle or not.

  • A𝐵𝐶=37.5, a right-angled triangle
  • B𝐵𝐶=40.2, not a right-angled triangle
  • C𝐵𝐶=35.4, a right-angled triangle
  • D𝐵𝐶=2.9, not a right-angled triangle

Q7:

In triangle 𝐴𝐵𝐶, let 𝐷 on 𝐵𝐶 be the foot of the altitude from 𝐴. If 𝐴𝐶=118.9, 𝐴𝐷=69.618, and 𝐵𝐷=50.94, is 𝐴𝐵𝐶 right angled at 𝐴?

  • Ano
  • Byes

Q8:

In triangle 𝐴𝐵𝐶, 𝐴𝐷 is perpendicular to 𝐵𝐶, 𝐷 lies between 𝐵 and 𝐶, 𝐵𝐷=8, 𝐶𝐷=2, and 𝐴𝐷=4. Is 𝐴𝐵𝐶 a right-angled triangle?

  • Ano
  • Byes

Q9:

What does (𝐴𝐶) equal to?

  • A(𝐶𝐵)−(𝐴𝐵)
  • B(𝐶𝐷)−(𝐴𝐷)
  • C𝐶𝐷−𝐷𝐵
  • D𝐶𝐵−𝐴𝐵

Q10:

In the figure shown, suppose that 𝐴𝐸=2𝐵𝐶 and 𝐵𝐷=8. Determine 𝐴𝐷 and 𝐸𝐷 rounded to the nearest hundredth, if necessary.

  • A𝐴𝐷=13.87cm, 𝐸𝐷=25.97cm
  • B𝐴𝐷=8.8cm, 𝐸𝐷=17.74cm
  • C𝐴𝐷=13.87cm, 𝐸𝐷=24.17cm
  • D𝐴𝐷=8.8cm, 𝐸𝐷=21.67cm

Q11:

Is △𝐸𝐴𝐷 a right-angled triangle at 𝐴?

  • Ano
  • Byes

Q12:

𝐴𝐵𝐶 is a triangle where 𝐴𝐵=3cm, 𝐵𝐶=4cm and 𝐴𝐶=5cm. Find the size of ∠𝐴𝐵𝐶.

Q13:

𝐴𝐵𝐶 is a triangle where 𝐴𝐵=𝐴𝐵=5cm, 𝐵𝐶=12cm and 𝐴𝐶=13cm. Find the size of ∠𝐴𝐵𝐶.

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 𝐴𝐵, 𝐴𝐶, and 𝐵𝐶.

  • A𝐴𝐵=4, 𝐴𝐶=2√2, 𝐵𝐶=2√2
  • B𝐴𝐵=2√2, 𝐴𝐶=4, 𝐵𝐶=4
  • C𝐴𝐵=4, 𝐴𝐶=2√2, 𝐵𝐶=4
  • D𝐴𝐵=2√2, 𝐴𝐶=2√2, 𝐵𝐶=2√2
  • E𝐴𝐵=2√2, 𝐴𝐶=2√2, 𝐵𝐶=4

Using the Pythagorean theorem, decide: is triangle 𝐴𝐵𝐶 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 𝐴𝐵, 𝐴𝐶, and 𝐵𝐶

  • A𝐴𝐵=√74, 𝐴𝐶=2√2, 𝐵𝐶=√74
  • B𝐴𝐵=2√2, 𝐴𝐶=√74, 𝐵𝐶=√74
  • C𝐴𝐵=2√2, 𝐴𝐶=√74, 𝐵𝐶=2
  • D𝐴𝐵=2√2, 𝐴𝐶=√74, 𝐵𝐶=2√2
  • E𝐴𝐵=2√2, 𝐴𝐶=2√2, 𝐵𝐶=√74

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

  • AYes
  • BNo

Hence, are the two lines perpendicular?

  • AYes
  • BNo

Q16:

In rectangle 𝐴𝐵𝐶𝐷, suppose 𝐴𝐸=8, 𝐷𝐸=2, and 𝐷𝐶=4. Is △𝐵𝐸𝐶 right angled?

  • Ano
  • Byes

Q17:

In the isosceles triangle 𝐴𝐵𝐶 below, 𝐴𝐷=36cm, 𝐵𝐶=54cm, and 𝐴𝐸=60cm. Is △𝐵𝐴𝐸 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 △𝐴𝐶𝐷 a right-angled triangle at 𝐶?

  • Ano
  • Byes

Q21:

Is △𝐴𝐶𝐷 a right-angled triangle at 𝐶?

  • 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 𝐴𝐵𝐶 point 𝐷 lies on 𝐵𝐶 and 𝐴𝐷⟂𝐵𝐶, 𝐴𝐶=10.5, 𝐴𝐷=6, and 𝐴𝐵=7.69. Find the length of 𝐵𝐶 to the nearest tenth, and then determine whether △𝐴𝐵𝐶 is a right-angled triangle or not.

  • A𝐵𝐶=12.5, a right-angled triangle
  • B𝐵𝐶=13.4, not a right-angled triangle
  • C𝐵𝐶=8.2, a right-angled triangle
  • D𝐵𝐶=4.4, not a right-angled triangle

Q25:

Is this triangle a right triangle?

  • AYes
  • BNo

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