Lesson Worksheet: The Converse of the Pythagorean Theorem Mathematics • 8th Grade

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 triangle?

  • ANo
  • BYes

Q3:

Is △𝐴𝐢𝐷 a right 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, 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 triangle or not.

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

This lesson includes 10 additional questions and 90 additional question variations for subscribers.

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