Let's solve the problem step by step:
### Step 1: Understanding the Given Data
- Triangle ABC is a right triangle with angle A being 90°.
- The side AB is [tex]\( x \)[/tex] cm.
- The side AC is [tex]\( x + 5 \)[/tex] cm.
- The area of the triangle is [tex]\( 150 \text{ cm}^2 \)[/tex].
### Step 2: Setting Up the Area Formula
For a right triangle with legs [tex]\( AB \)[/tex] and [tex]\( AC-AB \)[/tex]:
- The area of a right triangle is given by [tex]\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)[/tex].
Here, [tex]\( AB \)[/tex] is one leg (base) and [tex]\( AC - AB \)[/tex] is the other leg (height). Hence:
[tex]\[ \text{Area} = \frac{1}{2} \times AB \times (AC - AB) \][/tex]
### Step 3: Plugging In the Values
Given that the area is [tex]\( 150 \text{ cm}^2 \)[/tex]:
[tex]\[ 150 = \frac{1}{2} \times x \times (x + 5 - x) \][/tex]
This simplifies to:
[tex]\[ 150 = \frac{1}{2} \times x \times 5 \][/tex]
### Step 4: Solving for [tex]\( x \)[/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[ 300 = 5x \][/tex]
Divide both sides by 5:
[tex]\[ x = 60 \][/tex]
### Step 5: Finding the Lengths of the Sides
Now that we have [tex]\( x = 60 \)[/tex] cm:
- Side [tex]\( AB \)[/tex] (one leg) = [tex]\( x = 60 \)[/tex] cm.
- Side [tex]\( AC \)[/tex] (the hypotenuse) = [tex]\( x + 5 = 60 + 5 = 65 \)[/tex] cm.
To find the length of the hypotenuse [tex]\( BC \)[/tex], we can use the Pythagorean theorem, which states [tex]\( BC^2 = AB^2 + (AC - AB)^2 \)[/tex].
Here, [tex]\( BC \)[/tex] is the hypotenuse:
[tex]\[ BC = \sqrt{AB^2 + AC^2} \][/tex]
So:
[tex]\[ BC = \sqrt{60^2 + 65^2} \][/tex]
[tex]\[ BC = \sqrt{3600 + 4225} \][/tex]
[tex]\[ BC = \sqrt{7825} \][/tex]
[tex]\[ BC ≈ 88.46 \][/tex]
### Final Answer:
The sides of the triangle are:
- [tex]\( AB = 60 \)[/tex] cm
- [tex]\( AC = 65 \)[/tex] cm
- [tex]\( BC ≈ 88.46 \)[/tex] cm
