Table with Completed Lengths

Oct 13, 2025 by windailys.com 62 views

Introduction

In this activity, we will be working with a table that contains information about the lengths of sides of triangles. The table has three columns: $a$ , $b$ , and $c$ , which represent the lengths of the sides of the triangles. However, there are some missing lengths in the table that we need to complete. In this article, we will go through the process of completing the table by supplying the missing lengths.

Understanding the Table

The table has three rows, each representing a different triangle. The columns $a$ , $b$ , and $c$ represent the lengths of the sides of the triangles. The missing lengths are represented by empty cells in the table.

	$a$	$b$	$c$
1	4	15
2	5		$7 \sqrt{3}$
3	6

Completing the Table

To complete the table, we need to use the information given in the table to find the missing lengths. Let's start by analyzing the first row.

Row 1

In the first row, we have the lengths of sides $a$ and $b$ as 4 and 15, respectively. However, the length of side $c$ is missing. To find the length of side $c$ , we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Let's assume that the length of side $c$ is $x$ . Then, using the Pythagorean theorem, we can write:

x^2 = 4^2 + 15^2

Simplifying the equation, we get:

x^2 = 16 + 225

x^2 = 241

Taking the square root of both sides, we get:

x = \sqrt{241}

Therefore, the length of side $c$ in the first row is $\sqrt{241}$ .

Row 2

In the second row, we have the lengths of sides $a$ and $c$ as 5 and $7 \sqrt{3}$ , respectively. However, the length of side $b$ is missing. To find the length of side $b$ , we can use the Pythagorean theorem again.

Let's assume that the length of side $b$ is $y$ . Then, using the Pythagorean theorem, we can write:

y^2 = 5^2 + (7 \sqrt{3})^2

Simplifying the equation, we get:

y^2 = 25 + 147 \cdot 3

y^2 = 25 + 441

y^2 = 466

Taking the square root of both sides, we get:

y = \sqrt{466}

Therefore, the length of side $b$ in the second row is $\sqrt{466}$ .

Row 3

In the third row, we have the length of side $a$ as 6. However, the lengths of sides $b$ and $c$ are missing. To find the lengths of sides $b$ and $c$ , we can use the Pythagorean theorem again.

Let's assume that the length of side $b$ is $z$ . Then, using the Pythagorean theorem, we can write:

z^2 = 6^2 + x^2

where $x$ is the length of side $c$ . However, we don't know the value of $x$ yet. We can express $x$ in terms of $z$ and $y$ using the Pythagorean theorem:

x^2 = z^2 - 6^2

Substituting this expression into the previous equation, we get:

z^2 = 6^2 + (z^2 - 6^2)

Simplifying the equation, we get:

z^2 = z^2

This equation is true for all values of $z$ . Therefore, we can't find a unique value for $z$ using the Pythagorean theorem alone.

However, we can use the fact that the sum of the squares of the lengths of the sides of a triangle is equal to twice the square of the length of the altitude from the right angle to the hypotenuse. Let's call the length of the altitude $h$ . Then, we can write:

a^2 + b^2 = 2c^2

Substituting the values of $a$ , $b$ , and $c$ , we get:

6^2 + z^2 = 2x^2

Simplifying the equation, we get:

36 + z^2 = 2x^2

Substituting the expression for $x^2$ in terms of $z$ and $y$ , we get:

36 + z^2 = 2(z^2 - 36)

Simplifying the equation, we get:

36 + z^2 = 2z^2 - 72

Rearranging the equation, we get:

z^2 = 108

Taking the square root of both sides, we get:

z = \sqrt{108}

Therefore, the length of side $b$ in the third row is $\sqrt{108}$ .

Now that we have found the length of side $b$ , we can find the length of side $c$ using the Pythagorean theorem:

x^2 = z^2 - 6^2

Substituting the value of $z$ , we get:

x^2 = 108 - 36

x^2 = 72

Taking the square root of both sides, we get:

x = \sqrt{72}

Therefore, the length of side $c$ in the third row is $\sqrt{72}$ .

Conclusion

In this activity, we completed the table by supplying the missing lengths. We used the Pythagorean theorem to find the missing lengths in each row. In the first row, we found the length of side $c$ to be $\sqrt{241}$ . In the second row, we found the length of side $b$ to be $\sqrt{466}$ . In the third row, we found the lengths of sides $b$ and $c$ to be $\sqrt{108}$ and $\sqrt{72}$ , respectively.

Table with Completed Lengths

	$a$	$b$	$c$
1	4	15	$\sqrt{241}$
2	5	$\sqrt{466}$	$7 \sqrt{3}$
3	6	$\sqrt{108}$	$\sqrt{72}$

Discussion

This activity demonstrates the importance of using mathematical techniques to solve problems. In this case, we used the Pythagorean theorem to find the missing lengths in the table. The Pythagorean theorem is a fundamental concept in geometry that can be used to solve a wide range of problems.

The activity also highlights the importance of being able to express mathematical concepts in different ways. In this case, we used the Pythagorean theorem to find the missing lengths, but we also used the fact that the sum of the squares of the lengths of the sides of a triangle is equal to twice the square of the length of the altitude from the right angle to the hypotenuse.

Introduction

In the previous article, we completed a table with missing lengths by using the Pythagorean theorem. In this article, we will answer some common questions that may arise when working with the table.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Q: How do I use the Pythagorean theorem to find the missing lengths in the table?

A: To use the Pythagorean theorem to find the missing lengths in the table, you need to identify the lengths of the sides of the triangle and then use the formula:

c^2 = a^2 + b^2

where $c$ is the length of the hypotenuse, and $a$ and $b$ are the lengths of the other two sides.

Q: What if I don't know the length of one of the sides?

A: If you don't know the length of one of the sides, you can use the Pythagorean theorem to find it. For example, if you know the lengths of sides $a$ and $c$ , you can use the formula:

b^2 = c^2 - a^2

to find the length of side $b$ .

Q: Can I use the Pythagorean theorem to find the length of the altitude from the right angle to the hypotenuse?

A: Yes, you can use the Pythagorean theorem to find the length of the altitude from the right angle to the hypotenuse. The formula is:

h^2 = \frac{1}{2} (a^2 + b^2)

where $h$ is the length of the altitude, and $a$ and $b$ are the lengths of the other two sides.

Q: What if I make a mistake when using the Pythagorean theorem?

A: If you make a mistake when using the Pythagorean theorem, you may get an incorrect answer. To avoid this, make sure to double-check your calculations and use a calculator or other tool to verify your answer.

Q: Can I use the Pythagorean theorem to solve problems that involve non-right triangles?

A: No, the Pythagorean theorem only works for right triangles. If you are working with a non-right triangle, you will need to use a different formula or technique to find the length of the sides.

Q: Are there any other formulas or techniques that I can use to find the length of the sides of a triangle?

A: Yes, there are many other formulas and techniques that you can use to find the length of the sides of a triangle. Some examples include:

The Law of Cosines: This formula states that the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, minus twice the product of the lengths of the other two sides and the cosine of the angle between them.
The Law of Sines: This formula states that the ratio of the length of one side of a triangle to the sine of the angle opposite it is equal to the ratio of the length of another side to the sine of the angle opposite it.
Trigonometry: This branch of mathematics deals with the relationships between the sides and angles of triangles. It can be used to find the length of the sides of a triangle, as well as the angles between them.

Conclusion

In this article, we answered some common questions that may arise when working with the table. We also discussed some of the formulas and techniques that you can use to find the length of the sides of a triangle. Remember to always double-check your calculations and use a calculator or other tool to verify your answer.

Table with Completed Lengths

	$a$	$b$	$c$
1	4	15	$\sqrt{241}$
2	5	$\sqrt{466}$	$7 \sqrt{3}$
3	6	$\sqrt{108}$	$\sqrt{72}$

Discussion

This article demonstrates the importance of being able to answer common questions that may arise when working with mathematical concepts. It also highlights the importance of using a variety of formulas and techniques to solve problems. Remember to always double-check your calculations and use a calculator or other tool to verify your answer.