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Pythagorean theorem concept

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5364308
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) \(h^2+i^2=g^2\)
5365738
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 \(\overline{XZ}\).
5101338
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.
5101358
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 \(7^2+24^2=25^2\).
5149988
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.
5150148
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
5150158
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
5150208
A triangular garden bed has side lengths \(a=9\,\text{m}\), \(b=12\,\text{m}\), and \(c=16\,\text{m}\). a) Determine whether the corner between sides \(a\) and \(b\) is a right angle. b) Use a comparison of \(a^2+b^2\) and \(c^2\) to explain whether that angle is greater than or less than \(90^\circ\).

Hints

- Calculate the square of each side length. - Compare the sum of the squares of the two sides that form the corner with the square of the opposite side. - Consider what happens to an angle when its opposite side is longer than it would be in a right triangle.

Solution

1. The angle between sides \(a\) and \(b\) is opposite side \(c\). Since \(a^2+b^2=9^2+12^2=225\) and \(c^2=16^2=256\), the angle is not a right angle. 2. Because \(c^2>a^2+b^2\), the angle opposite side \(c\) is greater than \(90^\circ\).

Answer

a) No. The corner is not a right angle because \(9^2+12^2=225\ne256=16^2\). b) The angle is greater than \(90^\circ\) because \(c^2>a^2+b^2\).
5152448
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.
5155408
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 \(B\).
5280828
In a right triangle, the square built on one leg has area \(72\,\text{cm}^2\). The square built on the hypotenuse has area \(121\,\text{cm}^2\). a) Without first finding any side length, find the area of the square built on the other leg. Explain the relationship you used. b) Find the length of the other leg. c) Find the exact lengths of the given leg and the hypotenuse.

Hints

- Focus first on the areas of the three squares rather than their side lengths. - Which square corresponds to the hypotenuse of the right triangle? - After finding a square's area, how can you recover its side length?

Solution

1. For a right triangle, the areas of the squares on the two legs add to the area of the square on the hypotenuse. If the missing square area is \(A\), then \(72+A=121\), so \(A=49\,\text{cm}^2\). 2. The other leg is the side of a square with area \(49\,\text{cm}^2\), so its length is \(\sqrt{49}=7\,\text{cm}\). 3. The given leg has length \(\sqrt{72}=6\sqrt{2}\,\text{cm}\), and the hypotenuse has length \(\sqrt{121}=11\,\text{cm}\).

Answer

a) \(49\,\text{cm}^2\). The two leg-square areas add to the hypotenuse-square area. b) \(7\,\text{cm}\) c) The given leg is \(6\sqrt{2}\,\text{cm}\), and the hypotenuse is \(11\,\text{cm}\).
5364318
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: \(r^2+h^2=a^2\)
5101348
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.
5150168
A triangle has side lengths \(\sqrt{2}\,\text{m}\), \(\sqrt{7}\,\text{m}\), and \(300\,\text{cm}\). Determine whether it is a right triangle. If it is, find its exact area.

Hints

- Express all three side lengths in the same unit before comparing them. - Which side is longest after the conversion? - How can the three squared side lengths tell you whether the triangle is right? - If the triangle is right, which two sides can serve as the base and height?

Solution

1. Convert \(300\,\text{cm}\) to \(3\,\text{m}\). The longest side is \(3\,\text{m}\). 2. The squares of the two shorter side lengths sum to \((\sqrt{2})^2+(\sqrt{7})^2=2+7=9\), and the square of the longest side is \(3^2=9\). Therefore, the triangle is right. 3. The two shorter sides are the legs, so the area is \(\frac{1}{2}(\sqrt{2})(\sqrt{7})=\frac{\sqrt{14}}{2}\,\text{m}^2\).

Answer

The triangle is right, and its area is \(\frac{\sqrt{14}}{2}\,\text{m}^2\).
5152468
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.
5154448
A triangle has side lengths \(a=8\,\text{cm}\), \(b=15\,\text{cm}\), and \(c=17\,\text{cm}\). Side \(c\) is opposite \(\angle C\). a) Show with calculations that the triangle is right. b) Suppose side \(c\) is shortened to \(16\,\text{cm}\) while \(a\) and \(b\) stay the same. Use calculations to determine whether \(\angle C\) is now acute or obtuse.

Hints

- Use the converse of the Pythagorean theorem. - Identify the condition that gives a right angle. - Think about how shortening the side opposite an angle changes that angle when the other two sides stay fixed. - Compare \(c^2\) with \(a^2+b^2\).

Solution

1. \(8^2+15^2=64+225=289\), and \(17^2=289\). Because \(8^2+15^2=17^2\), the triangle is right and \(\angle C=90^\circ\). 2. After shortening \(c\), the sum \(a^2+b^2\) remains \(289\), while \(c^2=16^2=256\). Since \(c^2<a^2+b^2\), the angle opposite side \(c\) is acute, so \(\angle C<90^\circ\).

Answer

a) The triangle is right because \(8^2+15^2=17^2\). b) \(\angle C\) is acute because \(16^2<8^2+15^2\).
5512428
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 \(a^2+b^2=c^2\).
5371868
Triangle \(OAB\) is shown on the coordinate plane. a) Read the coordinates and find the lengths of all three sides. b) Use the side lengths to determine \(\angle AOB\). Justify your conclusion.
Figure for problem 537186

Hints

- Read the three coordinates from the graph before calculating. - Compare the squares of the side lengths to test for a right angle. - Equal side lengths imply equal opposite angles. - Use the angle sum of a triangle after identifying the right angle.

Solution

1. From the graph, \(O(0, 0)\), \(A(3, 1)\), and \(B(1, 2)\). 2. The side lengths are \(OA=\sqrt{3^2+1^2}=\sqrt{10}\), \(OB=\sqrt{1^2+2^2}=\sqrt{5}\), and \(AB=\sqrt{(3-1)^2+(1-2)^2}=\sqrt{5}\). 3. Since \(OB^2+AB^2=5+5=10=OA^2\), the triangle is right with the right angle at \(B\). 4. Also, \(OB=AB\), so the two acute angles are equal. They sum to \(90^\circ\), so \(\angle AOB=45^\circ\).

Answer

a) \(OA=\sqrt{10}\), \(OB=\sqrt{5}\), and \(AB=\sqrt{5}\) b) \(\angle AOB=45^\circ\)
5512438
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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