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Pythagorean theorem concept

Learning goals

  • Explains the relationship a² + b² = c² for right triangles.
  • Identifies the hypotenuse correctly.
  • Connects the theorem to areas of squares on triangle sides.
  • Distinguishes right-triangle use from non-right triangles.

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Pythagorean theorem concept 1
Write the Pythagorean equation for each right triangle shown.
Figure for problem 536430

Hints

- The hypotenuse is opposite the right angle. - The two sides that form the right angle are the legs. - Use the form “leg squared plus leg squared equals hypotenuse squared.”

Solution

1. In diagram a), the legs are \(u\) and \(v\), and the hypotenuse is \(w\). Therefore, \(u^2+v^2=w^2\). 2. In diagram b), the legs are \(y\) and \(z\), and the hypotenuse is \(x\). Therefore, \(y^2+z^2=x^2\). 3. In diagram c), the legs are \(h\) and \(i\), and the hypotenuse is \(g\). Therefore, \(h^2+i^2=g^2\).

Answer

a) \(u^2+v^2=w^2\) b) \(y^2+z^2=x^2\) c) \(h^2+i^2=g^2\)
Pythagorean theorem concept 2
In right triangle \(XYZ\), the right angle is at vertex \(Y\). Name the two legs and the hypotenuse.
Figure for problem 536573

Hints

- Locate the right angle. - Which two sides meet to form that angle? - What is the name of the side opposite the right angle?

Solution

1. The legs are the two sides that form the right angle. Since the right angle is at \(Y\), the legs are \(\overline{XY}\) and \(\overline{YZ}\). 2. The hypotenuse is the side opposite the right angle, so it is \(\overline{XZ}\).

Answer

The legs are \(\overline{XY}\) and \(\overline{YZ}\). The hypotenuse is \(\overline{XZ}\).
Pythagorean theorem concept 3
A right triangle has a square built on each side. The squares on its legs have areas \(9\,\text{cm}^2\) and \(16\,\text{cm}^2\). What is the area of the square on the hypotenuse?

Hints

- Which side of a right triangle is longest? - How might the two smaller square areas be related to the square on that side?

Solution

1. The Pythagorean theorem says the area of the hypotenuse square equals the sum of the areas of the two leg squares. 2. Add the areas: \(9+16=25\,\text{cm}^2\).

Answer

\(25\,\text{cm}^2\).
Pythagorean theorem concept 4
A model builder has three sets of wooden rods. Which sets can be used as the side lengths of a right triangle? Set 1: \(11\,\text{cm}, 60\,\text{cm}, 61\,\text{cm}\) Set 2: \(10\,\text{cm}, 24\,\text{cm}, 25\,\text{cm}\) Set 3: \(20\,\text{cm}, 21\,\text{cm}, 29\,\text{cm}\)

Hints

- Start with the longest rod in each set. - For a right triangle, the square of the longest side has a specific relationship to the squares of the other two sides. - Check each set separately instead of judging by how the numbers look. - Also make sure the three lengths can form a triangle.

Solution

1. Set 1: The longest side is \(61\,\text{cm}\). Since \(11^2 + 60^2 = 121 + 3600 = 3721\) and \(61^2 = 3721\), the side lengths form a right triangle. 2. Set 2: The longest side is \(25\,\text{cm}\). Since \(10^2 + 24^2 = 100 + 576 = 676\) but \(25^2 = 625\), the side lengths do not form a right triangle. 3. Set 3: The longest side is \(29\,\text{cm}\). Since \(20^2 + 21^2 = 400 + 441 = 841\) and \(29^2 = 841\), the side lengths form a right triangle.

Answer

Sets 1 and 3 form right triangles. Set 2 does not form a right triangle.
Pythagorean theorem concept 5
A triangular wooden brace has side lengths \(24\,\text{cm}\), \(7\,\text{cm}\), and \(25\,\text{cm}\). Use the converse of the Pythagorean theorem to determine whether the brace has an exact right angle.

Hints

- Order the three side lengths from least to greatest. - Decide which side would have to be the hypotenuse. - Compare the squares of the side lengths rather than comparing the lengths directly. - State clearly whether the brace has an exact right angle.

Solution

1. The longest side is \(25\,\text{cm}\), so it is the possible hypotenuse. 2. Compare the sum of the squares of the shorter sides with the square of the longest side: \(7^2+24^2=49+576=625\). 3. Since \(25^2=625\), the equality \(7^2+24^2=25^2\) holds. By the converse of the Pythagorean theorem, the brace has a right angle.

Answer

Yes. The brace has a right angle because \(7^2+24^2=25^2\).
Pythagorean theorem concept 6
Each equation represents the relationship among the side lengths of a right triangle. For each equation, identify the variable that represents the hypotenuse and briefly explain your reasoning. 1. \(p^2 = q^2 - r^2\) 2. \(c = \sqrt{a^2 + b^2}\) 3. \(x^2 + x^2 = y^2\) 4. \(s^2 - t^2 - u^2 = 0\)

Hints

- Recall the Pythagorean theorem in the form \(a^2 + b^2 = c^2\). Which side is represented by \(c\)? - Rewrite each equation so that one squared variable is alone on one side. - For the equation with a square root, consider what happens when both sides are squared.

Solution

1. Rewrite the equation as \(q^2 = p^2 + r^2\). The square of \(q\) equals the sum of the squares of the other two sides, so \(q\) is the hypotenuse. 2. Squaring both sides gives \(c^2 = a^2 + b^2\), so \(c\) is the hypotenuse. 3. The equation already has the form “leg squared plus leg squared equals hypotenuse squared,” so \(y\) is the hypotenuse. 4. Rewrite the equation as \(s^2 = t^2 + u^2\), so \(s\) is the hypotenuse.

Answer

1. \(q\) is the hypotenuse. 2. \(c\) is the hypotenuse. 3. \(y\) is the hypotenuse. 4. \(s\) is the hypotenuse.
Pythagorean theorem concept 7
You can classify a triangle by comparing the sum of the squares of the two shorter sides, \(a^2+b^2\), with the square of the longest side, \(c^2\). If \(a^2+b^2>c^2\), the triangle is acute. If \(a^2+b^2=c^2\), the triangle is right. If \(a^2+b^2<c^2\), the triangle is obtuse. Classify each triangle: a) \(7\,\text{cm}, 8\,\text{cm}, 12\,\text{cm}\) b) \(9\,\text{cm}, 40\,\text{cm}, 41\,\text{cm}\) c) \(6\,\text{cm}, 7\,\text{cm}, 9\,\text{cm}\)

Hints

- Identify the longest side in each triangle. - Calculate the square of each side length. - Compare the square of the longest side with the sum of the other two squares.

Solution

1. For part a, \(7^2+8^2=113\) and \(12^2=144\). Since \(113<144\), the triangle is obtuse. 2. For part b, \(9^2+40^2=1681\) and \(41^2=1681\). Since the values are equal, the triangle is right. 3. For part c, \(6^2+7^2=85\) and \(9^2=81\). Since \(85>81\), the triangle is acute.

Answer

a) Obtuse b) Right c) Acute
Pythagorean theorem concept 8
Use the converse of the Pythagorean theorem to determine whether each set of side lengths forms a right triangle: a) \(a=\sqrt{7}\), \(b=\sqrt{18}\), \(c=5\) b) \(a=3\), \(b=\sqrt{10}\), \(c=\sqrt{19}\) c) \(a=\sqrt{6}\), \(b=\sqrt{8}\), \(c=4\)

Hints

- Use the fact that squaring a square root gives the number under the radical. - Identify the longest side in each part by comparing the side lengths. - Check whether the sum of the squares of the two shorter sides equals the square of the longest side.

Solution

1. For part a, the squared side lengths are \(7\), \(18\), and \(25\). Since \(7+18=25\), the triangle is right. 2. For part b, the squared side lengths are \(9\), \(10\), and \(19\). Since \(9+10=19\), the triangle is right. 3. For part c, the squared side lengths are \(6\), \(8\), and \(16\). Since \(6+8=14\ne16\), the triangle is not right.

Answer

a) Right triangle b) Right triangle c) Not a right triangle
Pythagorean theorem concept 9
In \(\triangle RST\), the right angle is at vertex \(S\). Side \(r\) is opposite vertex \(R\), side \(s\) is opposite vertex \(S\), and side \(t\) is opposite vertex \(T\). a) Identify the legs and the hypotenuse. b) Write the Pythagorean theorem for this triangle. c) Explain why the formula \(a^2+b^2=c^2\) cannot be applied to every right triangle without first identifying the hypotenuse.

Hints

- Which side is opposite the right angle? - Which two sides meet to form the right angle? - Think about whether side letters have fixed roles in every diagram.

Solution

1. The hypotenuse is opposite the right angle. Because the right angle is at \(S\), side \(s\) is the hypotenuse. The legs are \(r\) and \(t\). 2. The Pythagorean equation is \(r^2+t^2=s^2\). 3. In \(a^2+b^2=c^2\), the letter \(c\) represents the hypotenuse by definition. Other diagrams may use different side labels, so the hypotenuse must be identified before assigning the variables in the formula.

Answer

a) The legs are \(r\) and \(t\); the hypotenuse is \(s\). b) \(r^2+t^2=s^2\) c) The form \(a^2+b^2=c^2\) works only when \(c\) has been defined as the hypotenuse.
Pythagorean theorem concept 10
In \(\triangle ABC\), sides \(a\), \(b\), and \(c\) are opposite vertices \(A\), \(B\), and \(C\), respectively, and \(a^2+c^2=b^2\). a) At which vertex, \(A\), \(B\), or \(C\), is the right angle? b) Suppose side \(a\) is made longer while side \(c\) stays fixed. If the triangle must remain right, what happens to the hypotenuse and the location of the right angle? Explain.

Hints

- Which side is alone as a square in the Pythagorean equation? - A side named with a lowercase letter lies opposite the matching uppercase vertex. - If one leg increases and the other stays fixed, what must happen to the hypotenuse for the equation to remain true?

Solution

1. In \(a^2+c^2=b^2\), side \(b\) is the hypotenuse because its square equals the sum of the squares of the other two sides. The right angle is opposite side \(b\), so it is at vertex \(B\). 2. The hypotenuse must satisfy \(b=\sqrt{a^2+c^2}\). As \(a\) increases while \(c\) stays fixed, \(b\) also increases. Sides \(a\) and \(c\) remain the legs, so the right angle stays at vertex \(B\).

Answer

a) The right angle is at vertex \(B\). b) The hypotenuse \(b\) becomes longer, and the right angle remains at vertex \(B\).
Pythagorean theorem concept 11
In a right triangle, the square on one leg has area \(72\,\text{cm}^2\). The square on the hypotenuse has area \(121\,\text{cm}^2\). a) Without first calculating lengths, find the area of the square on the other leg. Explain. b) Find the length of that other leg. c) Find the exact lengths of the given leg and the hypotenuse. You may leave square roots unsimplified.

Hints

- Relate the three square areas before taking any roots. - Use the area of a square to recover its side length. - How could you check that a length found from an area is reasonable?

Solution

1. The two leg-square areas sum to the hypotenuse-square area. Thus \(72+A=121\), giving \(A=49\,\text{cm}^2\). 2. The other leg has length \(\sqrt{49}=7\,\text{cm}\). 3. The given leg has length \(\sqrt{72}\,\text{cm}\); the hypotenuse is \(\sqrt{121}=11\,\text{cm}\).

Answer

a) \(49\,\text{cm}^2\), by subtracting the known leg-square area from the hypotenuse-square area. b) \(7\,\text{cm}\). c) Given leg: \(\sqrt{72}\,\text{cm}\); hypotenuse: \(11\,\text{cm}\).
Pythagorean theorem concept 12
The diagram shows a large triangle divided by altitude \(h\) into two right triangles. Write the Pythagorean equation for each smaller triangle.
Figure for problem 536431

Hints

- Locate the right angle in each smaller triangle. - Identify the legs and hypotenuse separately for each triangle. - Remember that side \(h\) belongs to both right triangles.

Solution

1. In the left triangle, the legs are \(s\) and \(h\), and the hypotenuse is \(b\). Therefore, \(s^2+h^2=b^2\). 2. In the right triangle, the legs are \(r\) and \(h\), and the hypotenuse is \(a\). Therefore, \(r^2+h^2=a^2\).

Answer

Left triangle: \(s^2+h^2=b^2\) Right triangle: \(r^2+h^2=a^2\)
Pythagorean theorem concept 13
Could the triangle have a right angle? Use the three side lengths, not the appearance of the drawing, to explain.
Figure for problem 559994

Hints

- Which is the longest side? It would be the hypotenuse if the triangle were right. - Compare its square with the sum of the other two squares.

Solution

1. The longest side is \(6\,\text{cm}\). 2. The squares on the other two sides have areas \(3^2+4^2=25\), but the longest side's square has area \(6^2=36\). 3. The two values are unequal, so the triangle is not right.

Answer

No. \(3^2+4^2=25\neq36=6^2\), so it does not satisfy the Pythagorean converse.
Pythagorean theorem concept 14
A triangle has side lengths \(5\), \(12\), and \(13\) units. Find the area of a square on each side. Do the two smaller squares together have the area of the largest square?

Hints

- Which of the three squares would have the largest area? - Think about the area of a square whose side length is \(s\). - What numerical test would support your answer to the second question?

Solution

1. The square areas are \(5^2=25\), \(12^2=144\), and \(13^2=169\) square units. 2. Since \(25+144=169\), the two smaller areas add to the largest.

Answer

\(25\), \(144\), and \(169\) square units; yes, \(25+144=169\).
Pythagorean theorem concept 15
Determine whether each set of side lengths forms a right triangle. Pay attention to the units and convert them as needed: a) \(a=0.9\,\text{m}\), \(b=40\,\text{cm}\), \(c=80\,\text{cm}\) b) \(x=1.2\,\text{cm}\), \(y=3.5\,\text{cm}\), \(z=3.7\,\text{cm}\)

Hints

- First, express all three side lengths in the same unit for each part. - Then identify the longest side. - Compare the square of the longest side with the sum of the squares of the two shorter sides. - Check each unit conversion carefully because a small conversion error can change the result.

Solution

1. For part a, convert all lengths to centimeters: \(a=90\,\text{cm}\), \(b=40\,\text{cm}\), and \(c=80\,\text{cm}\). The longest side is \(90\,\text{cm}\). 2. Since \(40^2+80^2=8000\) and \(90^2=8100\), the triangle is not a right triangle. 3. For part b, the longest side is \(3.7\,\text{cm}\). Since \(1.2^2+3.5^2=1.44+12.25=13.69\) and \(3.7^2=13.69\), the triangle is a right triangle.

Answer

a) Not a right triangle. b) Right triangle.
Pythagorean theorem concept 16
A triangle has side lengths \(\sqrt{2}\,\text{m}\), \(\sqrt{7}\,\text{m}\), and \(300\,\text{cm}\). Is it a right triangle? If so, give its area as an exact expression.

Hints

- Put all side lengths into the same unit. - Use the converse of the Pythagorean theorem with the longest side. - Once you know the right angle, which two lengths form the base and height?

Solution

1. Convert \(300\,\text{cm}=3\,\text{m}\). This is the longest side. 2. The two shorter side squares add to \(2+7=9\), which equals \(3^2\); therefore, it is a right triangle. 3. Its legs are \(\sqrt{2}\,\text{m}\) and \(\sqrt{7}\,\text{m}\), so the area is \(\frac{1}{2}\cdot\sqrt{2}\cdot\sqrt{7}\,\text{m}^2\).

Answer

Yes. Its area is exactly \(\frac{\sqrt{2}\cdot\sqrt{7}}{2}\,\text{m}^2\).
Pythagorean theorem concept 17
Two edges of a triangular garden meet at a corner. They are \(9\,\text{m}\) and \(12\,\text{m}\) long. The third edge, between their far ends, is \(16\,\text{m}\) long. a) How long would the third edge be if the corner were a right angle? b) Is the actual corner angle greater or less than \(90^\circ\)? Explain by thinking of the two edges opening like a hinge.

Hints

- Think of the two known edges as the legs of a right triangle for part a). - If you keep two joined sticks the same length, what happens to the gap between their far ends as you open them? - Compare the real gap with the right-angle gap.

Solution

1. With a right angle, the distance between the far ends would be \(\sqrt{9^2+12^2}=\sqrt{225}=15\,\text{m}\). 2. The actual distance is \(16\,\text{m}\), which is longer. 3. Keeping both edges fixed in length, opening the corner wider increases the distance between their far ends. The actual angle is therefore greater than \(90^\circ\) (obtuse).

Answer

a) \(15\,\text{m}\). b) Greater than \(90^\circ\). The far ends are farther apart than they would be at a right angle, so the corner is more open.
Pythagorean theorem concept 18
In \(\triangle ABC\), sides \(a\), \(b\), and \(c\) are opposite vertices \(A\), \(B\), and \(C\), respectively, and \(b^2=c^2-a^2\). a) Rewrite the equation without subtraction. Which angle is the right angle? b) Another triangle has side lengths \(x=5\), \(y=12\), and \(z=13\). Determine whether it is a right triangle. c) A claim says, “If all side lengths of a right triangle are doubled, the new triangle is not right because all the squares become four times as large.” Evaluate the claim.

Hints

- Rearrange the equation into the usual Pythagorean form. - The right angle is opposite the hypotenuse. - Check what happens when every term in a true equation is multiplied by the same factor.

Solution

1. Add \(a^2\) to both sides to obtain \(a^2+b^2=c^2\). Thus, \(c\) is the hypotenuse and \(\angle C\) is the right angle. 2. The longest side in the second triangle is \(13\). Since \(5^2+12^2=25+144=169=13^2\), the triangle is right. 3. If \(a^2+b^2=c^2\), then after doubling every side, \((2a)^2+(2b)^2=4(a^2+b^2)=4c^2=(2c)^2\). The new triangle is still right, so the claim is false.

Answer

a) \(a^2+b^2=c^2\), so \(\angle C\) is the right angle. b) Yes. The triangle is right because \(5^2+12^2=13^2\). c) The claim is false. Doubling all three sides preserves the Pythagorean relationship.
Pythagorean theorem concept 19
Two sticks, \(8\,\text{cm}\) and \(15\,\text{cm}\) long, meet at a corner. Their far ends are \(17\,\text{cm}\) apart. a) Show that the angle at the corner is a right angle. b) The sticks keep their lengths, but the distance between their far ends becomes \(16\,\text{cm}\). Is the new corner angle acute or obtuse? Explain what happens as the sticks move.

Hints

- Which side would be the hypotenuse in the first triangle? - Check whether the three lengths satisfy the right-triangle equation. - Imagine the sticks joined by a hinge. What must change to bring their far ends closer?

Solution

1. The lengths satisfy \(8^2+15^2=64+225=289=17^2\). By the converse of the Pythagorean theorem, the original corner is a right angle. 2. The two stick lengths are unchanged but their far ends move closer together, from \(17\,\text{cm}\) to \(16\,\text{cm}\). 3. The angle between the sticks must close from \(90^\circ\), so the new angle is less than \(90^\circ\) (acute).

Answer

a) \(8^2+15^2=17^2\), so the angle is right. b) Acute. As the far ends move closer together, the corner angle becomes smaller than \(90^\circ\).
Pythagorean theorem concept 20
The diagram shows a square with side length \(a+b\) partitioned into four congruent right triangles with legs \(a\) and \(b\), surrounding a central square with side length \(c\). Use the areas in the diagram to prove \(a^2+b^2=c^2\).
Figure for problem 551242

Hints

- Find the area of the entire outer square in one way. - Find the same area by adding the four congruent triangles and the central square. - Set the two area expressions equal. - Simplify only after the geometric area equation is established.

Solution

The area of the large square is \((a+b)^2\). The four right triangles have total area \(4\left(\frac{1}{2}ab\right)=2ab\), and the central square has area \(c^2\). Therefore, \((a+b)^2=2ab+c^2\). Expanding gives \(a^2+2ab+b^2=2ab+c^2\). Subtracting \(2ab\) from both sides gives \(a^2+b^2=c^2\).

Answer

\((a+b)^2=2ab+c^2\), so \(a^2+b^2=c^2\).
Pythagorean theorem concept 21
Each grid cell is \(1\times1\) unit. Explain why the uncovered areas in a) and b) are equal. Then find the length of a triangle's hypotenuse.
Figure for problem 559996

Hints

- What has the same area in both drawings? - If you remove equal triangle areas from equal squares, what remains? - How can you name the sides of the uncovered squares using the triangle's sides?

Solution

1. Both large squares have the same area, and the four triangles in each have the same combined area. Therefore, the area left over in each diagram is equal. 2. In b), the uncovered squares have sides \(3\) and \(4\) units, so their total area is \(3^2+4^2=25\) square units. 3. In a), the uncovered square has each side equal to a triangle's hypotenuse \(c\). Thus \(c^2=3^2+4^2=25\), giving \(c=5\) units.

Answer

The uncovered areas are equal because equal areas of four identical triangles are removed from equal large squares. Thus \(c^2=3^2+4^2=25\), and the hypotenuse is \(5\) units.
Pythagorean theorem concept 22
Suppose a triangle has side lengths \(a\), \(b\), and \(c\), with \(c\) the longest side, and \(a^2+b^2=c^2\). Explain why the angle opposite side \(c\) must be a right angle.

Hints

- Compare the given triangle with a right triangle having the same two shorter side lengths. - Apply the Pythagorean theorem only to the constructed right triangle. - What does the equation tell you about the two hypotenuse lengths? - If two triangles have the same three side lengths, what can you conclude about corresponding angles?

Solution

Construct a right triangle with legs \(a\) and \(b\), and call its hypotenuse \(d\). By the Pythagorean theorem, \(d^2=a^2+b^2\). The given triangle satisfies \(a^2+b^2=c^2\), so \(d^2=c^2\). Since side lengths are positive, \(d=c\). The constructed right triangle and the original triangle therefore have the same three side lengths, so they are congruent. Thus the angle opposite \(c\) in the original triangle equals the right angle opposite \(d\) in the constructed triangle.

Answer

The angle opposite \(c\) is \(90^\circ\); this proves the converse of the Pythagorean theorem.

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