The Area Of A Right-angled Isosceles Triangle Is $4 , \text{cm}^2$. Work Out The Perimeter Of The Triangle In Centimetres. Give Your Answer In The Form $a + B \sqrt{c}$, Where $ A , B , A, B, A , B , [/tex] And $c$

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Introduction

In this article, we will explore the concept of a right-angled isosceles triangle and how to find its perimeter given its area. A right-angled isosceles triangle is a triangle with one right angle (90 degrees) and two sides of equal length. The area of a triangle is given by the formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$. We will use this formula to find the perimeter of the triangle.

The Formula for the Area of a Triangle

The formula for the area of a triangle is given by:

Area=12ร—baseร—height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

In the case of a right-angled isosceles triangle, the base and height are equal, so we can write:

Area=12ร—xร—x=x22\text{Area} = \frac{1}{2} \times x \times x = \frac{x^2}{2}

where $x$ is the length of the equal sides.

Given the Area of the Triangle

We are given that the area of the triangle is $4 , \text{cm}^2$. We can set up an equation using the formula for the area:

x22=4\frac{x^2}{2} = 4

Solving for x

To solve for $x$, we can multiply both sides of the equation by 2:

x2=8x^2 = 8

Taking the square root of both sides, we get:

x=8=22x = \sqrt{8} = 2\sqrt{2}

Finding the Perimeter of the Triangle

The perimeter of a triangle is the sum of the lengths of its sides. In the case of a right-angled isosceles triangle, the perimeter is given by:

Perimeter=x+x+x2+x2\text{Perimeter} = x + x + \sqrt{x^2 + x^2}

Substituting the value of $x$, we get:

Perimeter=22+22+(22)2+(22)2\text{Perimeter} = 2\sqrt{2} + 2\sqrt{2} + \sqrt{(2\sqrt{2})^2 + (2\sqrt{2})^2}

Simplifying the expression, we get:

Perimeter=42+8+8\text{Perimeter} = 4\sqrt{2} + \sqrt{8 + 8}

Perimeter=42+16\text{Perimeter} = 4\sqrt{2} + \sqrt{16}

Perimeter=42+4\text{Perimeter} = 4\sqrt{2} + 4

Conclusion

In this article, we have found the perimeter of a right-angled isosceles triangle given its area. We used the formula for the area of a triangle and solved for the length of the equal sides. We then used this value to find the perimeter of the triangle. The perimeter of the triangle is given by $4 + 4\sqrt{2}$.

Final Answer

The final answer is $4 + 4\sqrt{2}$.

References

  • [1] "Triangle" by Wikipedia. Retrieved 2023-12-01.
  • [2] "Right-angled isosceles triangle" by Math Open Reference. Retrieved 2023-12-01.

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Q: What is a right-angled isosceles triangle?

A: A right-angled isosceles triangle is a triangle with one right angle (90 degrees) and two sides of equal length. This means that the two shorter sides (the legs) are equal in length, and the longest side (the hypotenuse) is opposite the right angle.

Q: What is the formula for the area of a right-angled isosceles triangle?

A: The formula for the area of a triangle is given by:

Area=12ร—baseร—height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

In the case of a right-angled isosceles triangle, the base and height are equal, so we can write:

Area=12ร—xร—x=x22\text{Area} = \frac{1}{2} \times x \times x = \frac{x^2}{2}

where $x$ is the length of the equal sides.

Q: How do I find the perimeter of a right-angled isosceles triangle?

A: The perimeter of a triangle is the sum of the lengths of its sides. In the case of a right-angled isosceles triangle, the perimeter is given by:

Perimeter=x+x+x2+x2\text{Perimeter} = x + x + \sqrt{x^2 + x^2}

Substituting the value of $x$, we get:

Perimeter=42+4\text{Perimeter} = 4\sqrt{2} + 4

Q: What is the relationship between the area and the perimeter of a right-angled isosceles triangle?

A: The area of a right-angled isosceles triangle is given by $\frac{x^2}{2}$, and the perimeter is given by $4\sqrt{2} + 4$. We can see that the area is proportional to the square of the length of the equal sides, while the perimeter is proportional to the square root of the length of the equal sides.

Q: Can I use the Pythagorean theorem to find the length of the hypotenuse of a right-angled isosceles triangle?

A: Yes, you can use the Pythagorean theorem to find the length of the hypotenuse of a right-angled isosceles triangle. The Pythagorean theorem states that:

a2+b2=c2a^2 + b^2 = c^2

where $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse. In the case of a right-angled isosceles triangle, $a = b$, so we can write:

a2+a2=c2a^2 + a^2 = c^2

2a2=c22a^2 = c^2

c=2ac = \sqrt{2}a

Q: What is the relationship between the length of the equal sides and the length of the hypotenuse of a right-angled isosceles triangle?

A: The length of the hypotenuse of a right-angled isosceles triangle is given by $\sqrt{2}a$, where $a$ is the length of the equal sides. This means that the length of the hypotenuse is equal to the length of the equal sides multiplied by the square root of 2.

Q: Can I use the formula for the area of a triangle to find the length of the equal sides of a right-angled isosceles triangle?

A: Yes, you can use the formula for the area of a triangle to find the length of the equal sides of a right-angled isosceles triangle. The formula for the area of a triangle is given by:

Area=12ร—baseร—height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

In the case of a right-angled isosceles triangle, the base and height are equal, so we can write:

Area=12ร—xร—x=x22\text{Area} = \frac{1}{2} \times x \times x = \frac{x^2}{2}

where $x$ is the length of the equal sides.

Q: What is the relationship between the length of the equal sides and the area of a right-angled isosceles triangle?

A: The area of a right-angled isosceles triangle is given by $\frac{x^2}{2}$, where $x$ is the length of the equal sides. This means that the area is proportional to the square of the length of the equal sides.

Q: Can I use the formula for the perimeter of a triangle to find the length of the equal sides of a right-angled isosceles triangle?

A: Yes, you can use the formula for the perimeter of a triangle to find the length of the equal sides of a right-angled isosceles triangle. The formula for the perimeter of a triangle is given by:

Perimeter=x+x+x2+x2\text{Perimeter} = x + x + \sqrt{x^2 + x^2}

Substituting the value of $x$, we get:

Perimeter=42+4\text{Perimeter} = 4\sqrt{2} + 4

Q: What is the relationship between the length of the equal sides and the perimeter of a right-angled isosceles triangle?

A: The perimeter of a right-angled isosceles triangle is given by $4\sqrt{2} + 4$, where $x$ is the length of the equal sides. This means that the perimeter is proportional to the square root of the length of the equal sides.

Q: Can I use the Pythagorean theorem to find the length of the hypotenuse of a right-angled isosceles triangle?

A: Yes, you can use the Pythagorean theorem to find the length of the hypotenuse of a right-angled isosceles triangle. The Pythagorean theorem states that:

a2+b2=c2a^2 + b^2 = c^2

where $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse. In the case of a right-angled isosceles triangle, $a = b$, so we can write:

a2+a2=c2a^2 + a^2 = c^2

2a2=c22a^2 = c^2

c=2ac = \sqrt{2}a

Q: What is the relationship between the length of the equal sides and the length of the hypotenuse of a right-angled isosceles triangle?

A: The length of the hypotenuse of a right-angled isosceles triangle is given by $\sqrt{2}a$, where $a$ is the length of the equal sides. This means that the length of the hypotenuse is equal to the length of the equal sides multiplied by the square root of 2.

Q: Can I use the formula for the area of a triangle to find the length of the equal sides of a right-angled isosceles triangle?

A: Yes, you can use the formula for the area of a triangle to find the length of the equal sides of a right-angled isosceles triangle. The formula for the area of a triangle is given by:

Area=12ร—baseร—height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

In the case of a right-angled isosceles triangle, the base and height are equal, so we can write:

Area=12ร—xร—x=x22\text{Area} = \frac{1}{2} \times x \times x = \frac{x^2}{2}

where $x$ is the length of the equal sides.

Q: What is the relationship between the length of the equal sides and the area of a right-angled isosceles triangle?

A: The area of a right-angled isosceles triangle is given by $\frac{x^2}{2}$, where $x$ is the length of the equal sides. This means that the area is proportional to the square of the length of the equal sides.

Q: Can I use the formula for the perimeter of a triangle to find the length of the equal sides of a right-angled isosceles triangle?

A: Yes, you can use the formula for the perimeter of a triangle to find the length of the equal sides of a right-angled isosceles triangle. The formula for the perimeter of a triangle is given by:

Perimeter=x+x+x2+x2\text{Perimeter} = x + x + \sqrt{x^2 + x^2}

Substituting the value of $x$, we get:

Perimeter=42+4\text{Perimeter} = 4\sqrt{2} + 4

Q: What is the relationship between the length of the equal sides and the perimeter of a right-angled isosceles triangle?

A: The perimeter of a right-angled isosceles triangle is given by $4\sqrt{2} + 4$, where $x$ is the length of the equal sides. This means that the perimeter is proportional to the square root of the length of the equal sides.

Q: Can I use the Pythagorean theorem to find the length of the hypotenuse of a right-angled isosceles triangle