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 △𝐴𝐢𝐷 a right-angled triangle at 𝐢?

  • 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=π‘Ž+π‘οŠ¨οŠ¨οŠ¨ is necessarily a right triangle?

Let us assume that △𝐴𝐡𝐢 is of side lengths π‘Ž, 𝑏, and 𝑐, with 𝑐=π‘Ž+π‘οŠ¨οŠ¨οŠ¨. Let △𝐷𝐡𝐢 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?

  • Aπ‘Ž=𝑑+π‘οŠ¨οŠ¨οŠ¨
  • B𝑏=π‘Ž+π‘‘οŠ¨οŠ¨οŠ¨
  • C𝑑=π‘Ž+π‘οŠ¨οŠ¨οŠ¨

We know that for △𝐴𝐡𝐢, 𝑐=π‘Ž+π‘οŠ¨οŠ¨οŠ¨.

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 △𝐴𝐡𝐢?

  • AIt is similar to △𝐷𝐡𝐢, so it has a right angle at 𝐴.
  • BIt is congruent to △𝐷𝐡𝐢, so it has a right angle at 𝐢.
  • CIt is congruent to △𝐷𝐡𝐢, so it has a right angle at 𝐡.
  • DIt is congruent to △𝐷𝐡𝐢, so it has a right angle at 𝐴.
  • EIt is similar to △𝐷𝐡𝐢, so it has a right angle at 𝐢.

Q6:

In triangle 𝐴𝐡𝐢 point 𝐷 lies on 𝐡𝐢 and οƒ«π΄π·βŸ‚π΅πΆ, 𝐴𝐢=37.8, 𝐴𝐷=10.08, and 𝐴𝐡=10.76. Find the length of 𝐡𝐢 to the nearest tenth, and then determine whether △𝐴𝐡𝐢 is a right-angled triangle or not.

  • A𝐡𝐢=37.5, a right-angled triangle
  • B𝐡𝐢=40.2, not a right-angled triangle
  • C𝐡𝐢=35.4, a right-angled triangle
  • D𝐡𝐢=2.9, not a right-angled triangle

Q7:

In triangle 𝐴𝐡𝐢, let 𝐷 on 𝐡𝐢 be the foot of the altitude from 𝐴. If 𝐴𝐢=118.9, 𝐴𝐷=69.618, and 𝐡𝐷=50.94, is 𝐴𝐡𝐢 right angled at 𝐴?

  • Ano
  • Byes

Q8:

In triangle 𝐴𝐡𝐢, 𝐴𝐷 is perpendicular to 𝐡𝐢, 𝐷 lies between 𝐡 and 𝐢, 𝐡𝐷=8, 𝐢𝐷=2, and 𝐴𝐷=4. Is 𝐴𝐡𝐢 a right-angled triangle?

  • Ano
  • Byes

Q9:

What does (𝐴𝐢) equal to?

  • A(𝐢𝐡)βˆ’(𝐴𝐡)
  • B(𝐢𝐷)βˆ’(𝐴𝐷)
  • CπΆπ·βˆ’π·π΅
  • DπΆπ΅βˆ’π΄π΅

Q10:

In the figure shown, suppose that 𝐴𝐸=2𝐡𝐢 and 𝐡𝐷=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:

𝐴𝐡𝐢 is a triangle where 𝐴𝐡=3cm, 𝐡𝐢=4cm and 𝐴𝐢=5cm. Find the size of ∠𝐴𝐡𝐢.

Q13:

𝐴𝐡𝐢 is a triangle where 𝐴𝐡=𝐴𝐡=5cm, 𝐡𝐢=12cm and 𝐴𝐢=13cm. Find the size of ∠𝐴𝐡𝐢.

Q14:

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 𝐴𝐡, 𝐴𝐢, and 𝐡𝐢.

  • A𝐴𝐡=4, 𝐴𝐢=2√2, 𝐡𝐢=2√2
  • B𝐴𝐡=2√2, 𝐴𝐢=4, 𝐡𝐢=4
  • C𝐴𝐡=4, 𝐴𝐢=2√2, 𝐡𝐢=4
  • D𝐴𝐡=2√2, 𝐴𝐢=2√2, 𝐡𝐢=2√2
  • E𝐴𝐡=2√2, 𝐴𝐢=2√2, 𝐡𝐢=4

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

  • AYes
  • BNo

Are the two lines perpendicular?

  • AYes
  • BNo

Q15:

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 𝐴𝐡, 𝐴𝐢, and 𝐡𝐢

  • A𝐴𝐡=√74, 𝐴𝐢=2√2, 𝐡𝐢=√74
  • B𝐴𝐡=2√2, 𝐴𝐢=√74, 𝐡𝐢=√74
  • C𝐴𝐡=2√2, 𝐴𝐢=√74, 𝐡𝐢=2
  • D𝐴𝐡=2√2, 𝐴𝐢=√74, 𝐡𝐢=2√2
  • E𝐴𝐡=2√2, 𝐴𝐢=2√2, 𝐡𝐢=√74

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

  • AYes
  • BNo

Hence, are the two lines perpendicular?

  • AYes
  • BNo

Q16:

In rectangle 𝐴𝐡𝐢𝐷, suppose 𝐴𝐸=8, 𝐷𝐸=2, and 𝐷𝐢=4. Is △𝐡𝐸𝐢 right angled?

  • Ano
  • Byes

Q17:

In the isosceles triangle 𝐴𝐡𝐢 below, 𝐴𝐷=36cm, 𝐡𝐢=54cm, and 𝐴𝐸=60cm. Is △𝐡𝐴𝐸 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 △𝐴𝐢𝐷 a right-angled triangle at 𝐢?

  • Ano
  • Byes

Q21:

Is △𝐴𝐢𝐷 a right-angled triangle at 𝐢?

  • 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 𝐴𝐡𝐢 point 𝐷 lies on 𝐡𝐢 and οƒ«π΄π·βŸ‚π΅πΆ, 𝐴𝐢=10.5, 𝐴𝐷=6, and 𝐴𝐡=7.69. Find the length of 𝐡𝐢 to the nearest tenth, and then determine whether △𝐴𝐡𝐢 is a right-angled triangle or not.

  • A𝐡𝐢=12.5, a right-angled triangle
  • B𝐡𝐢=13.4, not a right-angled triangle
  • C𝐡𝐢=8.2, a right-angled triangle
  • D𝐡𝐢=4.4, not a right-angled triangle

Q25:

Is this triangle a right triangle?

  • AYes
  • BNo

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