The Ratio of the Hypotenuse to One of Its Legs

Oct 13, 2025 by talynt.org 59 views

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Introduction

In a right-angled triangle, the relationship between the lengths of the sides is governed by the Pythagorean theorem. However, when we are given the ratio of the hypotenuse to one of the legs, we can use this information to find the lengths of the other sides. In this article, we will explore how to find the perimeter of a right triangle when the ratio of the hypotenuse to one of its legs is given, along with the length of the other leg.

The Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this can be expressed as:

c^2 = a^2 + b^2

The Ratio of the Hypotenuse to One of Its Legs

In this problem, we are given that the ratio of the hypotenuse (c) to one of its legs (a) is $\frac{13}{12}$ . We can express this as:

c/a = 13/12

The Length of the Other Leg

We are also given that the length of the other leg (b) is 15 cm.

Finding the Length of the Hypotenuse

Using the ratio of the hypotenuse to one of its legs, we can set up an equation to find the length of the hypotenuse (c). We can multiply both sides of the equation by a to get:

c = (13/12)a

Finding the Length of One of the Legs

We can use the Pythagorean theorem to find the length of one of the legs (a). We can substitute the length of the hypotenuse (c) from the previous equation into the Pythagorean theorem:

((13/12)a)^2 = a^2 + 15^2

Simplifying the Equation

We can simplify the equation by expanding the left-hand side:

(169/144)a^2 = a^2 + 225

Rearranging the Equation

We can rearrange the equation to isolate the term with a^2:

(169/144 - 1)a^2 = 225

Simplifying the Equation

We can simplify the equation by combining the fractions:

(25/144)a^2 = 225

Multiplying Both Sides by 144

We can multiply both sides of the equation by 144 to eliminate the fraction:

25a^2 = 32400

Dividing Both Sides by 25

We can divide both sides of the equation by 25 to isolate a^2:

a^2 = 1296

Taking the Square Root of Both Sides

We can take the square root of both sides of the equation to find the length of one of the legs (a):

a = 36

Finding the Length of the Hypotenuse

We can use the ratio of the hypotenuse to one of its legs to find the length of the hypotenuse (c):

c = (13/12)a c = (13/12)(36) c = 39

Finding the Perimeter of the Triangle

The perimeter of a triangle is the sum of the lengths of its sides. We can find the perimeter of the triangle by adding the lengths of the three sides:

Perimeter = a + b + c Perimeter = 36 + 15 + 39 Perimeter = 90

Conclusion

In this article, we have shown how to find the perimeter of a right triangle when the ratio of the hypotenuse to one of its legs is given, along with the length of the other leg. We used the Pythagorean theorem and the ratio of the hypotenuse to one of its legs to find the lengths of the other sides, and then calculated the perimeter of the triangle. The perimeter of the triangle is 90 cm.

Final Answer

The final answer is: $\boxed{90}$

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Introduction

In our previous article, we explored how to find the perimeter of a right triangle when the ratio of the hypotenuse to one of its legs is given, along with the length of the other leg. In this article, we will answer some of the most frequently asked questions related to this topic.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that describes the relationship between the lengths of the sides of a right-angled triangle. It states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this can be expressed as:

c^2 = a^2 + b^2

Q: How do I find the length of the hypotenuse when the ratio of the hypotenuse to one of its legs is given?

A: To find the length of the hypotenuse, you can use the ratio of the hypotenuse to one of its legs. Let's say the ratio is c/a = 13/12. You can set up an equation to find the length of the hypotenuse (c):

c = (13/12)a

Q: How do I find the length of one of the legs when the ratio of the hypotenuse to one of its legs is given?

A: To find the length of one of the legs, you can use the Pythagorean theorem. Let's say the length of the hypotenuse (c) is 39 cm and the length of the other leg (b) is 15 cm. You can set up an equation to find the length of one of the legs (a):

((13/12)a)^2 = a^2 + 15^2

Q: What is the perimeter of a triangle?

A: The perimeter of a triangle is the sum of the lengths of its sides. To find the perimeter of a triangle, you can add the lengths of the three sides:

Perimeter = a + b + c

Q: How do I find the perimeter of a right triangle when the ratio of the hypotenuse to one of its legs is given?

A: To find the perimeter of a right triangle, you can use the ratio of the hypotenuse to one of its legs and the length of the other leg. Let's say the ratio of the hypotenuse to one of its legs is c/a = 13/12 and the length of the other leg (b) is 15 cm. You can find the length of the hypotenuse (c) and then calculate the perimeter of the triangle:

Perimeter = a + b + c Perimeter = 36 + 15 + 39 Perimeter = 90

Q: What are some common mistakes to avoid when working with the Pythagorean theorem?

A: Some common mistakes to avoid when working with the Pythagorean theorem include:

Not using the correct formula: c^2 = a^2 + b^2
Not squaring the lengths of the sides: c^2 = (a^2) + (b^2)
Not using the correct units: make sure to use the same units for all the lengths
Not checking the signs: make sure to check the signs of the lengths and the result

Q: How do I apply the Pythagorean theorem in real-life situations?

A: The Pythagorean theorem has many real-life applications, including:

Building design: architects use the Pythagorean theorem to calculate the lengths of walls and roofs
Engineering: engineers use the Pythagorean theorem to calculate the lengths of pipes and wires
Surveying: surveyors use the Pythagorean theorem to calculate the lengths of distances between landmarks
Physics: physicists use the Pythagorean theorem to calculate the lengths of distances between objects in motion

Conclusion

In this article, we have answered some of the most frequently asked questions related to the ratio between the hypotenuse and one of its legs in a right triangle. We have covered topics such as the Pythagorean theorem, finding the length of the hypotenuse, finding the length of one of the legs, and applying the Pythagorean theorem in real-life situations. We hope that this article has been helpful in clarifying any doubts you may have had about this topic.

Final Answer

The final answer is: $\boxed{90}$